【Java与数学】若不等式x^2-a*x+a<0的解集中恰有3个整数，求a的范围？

【分析】

既然不等式存在解集，说明一元二次方程x^2-a*x+a=0有解；

解之间横跨三个整数，说明Δ大于0；

三个整数必然是连续的，因为f(x)=x^2-a*x+a最多只与x轴存在两个交点，这是题设里的隐含条件。

【传统解法】

因为Δ>0,有a^2-4a>0,得到a∈(-∞,0)∪(4,+∞);

下面就这两段进行讨论。

（1）当a∈(-∞,0)即a<0时

若x取0，0^2-a*0+a=a<0,符合x^2-a*x+a<0，这说明3个连续整数中有一个是0；

若x取1，1^2-a*1+a=1>0,不符合x^2-a*x+a<0，这说明3个连续整数不会在0的右边，只可以向左走；

由于0是最右边的一个数，故三个连续整数是-2，-1，0；

由于f(x)=x^2-a*x+a开口向上，从图线上看，抛物线与x轴的交点一个在（0，1)之间，一个在（-3，-2）之间；

当交点在（0，1)之间时，必有f(1)>0,f(0)<0;

而f(1)=1-a+a=1>0,f(0)=0-0+a<0,这两个值是符合条件f(1)>0,f(0)<0的；

当交点在（-3，-2）之间，必有f(-3)>0,f(-2)<0;

而f(-3)=9+3a+a=9+4a>0,得到a>-9/4;

f(-2)=4+2a+a=4+3a<0,得到a<-4/3;

故a<0时，范围缩小为（-9/4，-4/3）

（2）当a∈(4,+∞)即a>4时

若x取0，f(0)=a>4>0,不符合x^2-a*x+a<0，说明0不是三整数之一；

若x取1，f(1)=1>0,不符合x^2-a*x+a<0，说明1也不是三整数之一；

若x取2，f(0)=4-a<0,符合x^2-a*x+a<0，说明2是三整数之一；

既然一直在向右走，三整数又是连续的，2是打头的数，故连续三整数是2，3，4；

由于f(x)=x^2-a*x+a开口向上，从图线上看，抛物线与x轴的交点一个在（1，2)之间，一个在（4，5）之间；

当交点在（1，2)之间，f(1)=1-a+a=1>0,f(2)=4-a<0,这两个值是符合条件的；

当交点在（4，5）之间，

f(4)=16-4a+a=16-3a<0,得到a>16/3;

f(5)=25-5a+a=25-4a>0,得到a<25/4;

故a>4时，范围缩小为（16/3，25/4）

最后，a的范围就是（-9/4，-4/3）∪（16/3，25/4）。

【得不到正确结论的传统解法】

方程x^2-a*x+a=0的右根是(a+(a^2-4*a)^0.5)/2,左根是(a-(a^2-4*a)^0.5)/2；

两根之间有三个连续整数，故2<(a^2-4*a)^0.5<3;

解此不等式得到a∈（-1.6，-0.828）∪（4.828，5.6）。

从中取值如5代入a，发现方程x^2-5x+5=0的两根间只有两个整数，说明此法是不正确的。

此法错在只是判断两根间距离在2到3个标准长度之间，并不是正好卡三个整数。因此结果不正确。

【程序思路】

 程序解此题几乎可以直译。

首先用求根公式得到两个根；

其次设定a从-100变化到+100，步长为0.001，分别得到真实的两根值；

再取两根间的整数，如果是3个就符合条件了。

【Java代码】

import java.io.BufferedWriter;
import java.io.FileOutputStream;
import java.io.OutputStreamWriter;
import java.util.ArrayList;
import java.util.List;

/**
 * 若不等式x^2-a*x+a<0的解集中恰有3个整数，求a的范围？
 * @author ufo
 *
 */
public class Test {
    public static void main(String[] args) throws Exception{
        StringBuilder sb=new StringBuilder();
        double a;
        for(a=-100;a<100;a+=0.001) {
            double min=(a-Math.sqrt(a*a-4*a))/2;
            double max=(a+Math.sqrt(a*a-4*a))/2;
            
            int realMin=(int)Math.ceil(min);// 大于min的最小整数
            int realMax=(int)Math.floor(max);// 小于max最大整数
            
            List ls=new ArrayList<Integer>();
            for(int i=realMin;i<=realMax;i++){
                ls.add(i);
            }
            
            if(ls.size()==3) {
                System.out.println("a="+a+" min="+min+" min="+max+" ls="+ls);
                sb.append("a="+a+" min="+min+" min="+max+" ls="+ls+"\n");
            }
        }
        
        FileOutputStream writerStream = new FileOutputStream("240331.txt");
        BufferedWriter writer = new BufferedWriter(new OutputStreamWriter(writerStream, "UTF-8"));
        writer.write(sb.toString());
        writer.close();
    }
}

从上看出，代码很简明，考虑到控制台输出有限，特地把结果输入到了文件中。

【结果文件240331.txt内容】

a=-2.2499999998864255 min=-2.9999999998788542 min=0.7499999999924285 ls=[-2, -1, 0]
a=-2.2489999998864256 min=-2.9989333142434873 min=0.7499333143570615 ls=[-2, -1, 0]
a=-2.2479999998864257 min=-2.997866590647783 min=0.7498665907613571 ls=[-2, -1, 0]
a=-2.246999999886426 min=-2.996799829057289 min=0.7497998291708627 ls=[-2, -1, 0]
a=-2.245999999886426 min=-2.995733029437509 min=0.7497330295510827 ls=[-2, -1, 0]
a=-2.244999999886426 min=-2.9946661917539026 min=0.7496661918674765 ls=[-2, -1, 0]
a=-2.243999999886426 min=-2.9935993159718857 min=0.7495993160854597 ls=[-2, -1, 0]
a=-2.2429999998864263 min=-2.9925324020568302 min=0.7495324021704042 ls=[-2, -1, 0]
a=-2.2419999998864264 min=-2.991465449974063 min=0.7494654500876368 ls=[-2, -1, 0]
a=-2.2409999998864265 min=-2.990398459688867 min=0.7493984598024404 ls=[-2, -1, 0]
a=-2.2399999998864266 min=-2.9893314311664794 min=0.7493314312800528 ls=[-2, -1, 0]
a=-2.2389999998864267 min=-2.988264364372095 min=0.7492643644856682 ls=[-2, -1, 0]
a=-2.237999999886427 min=-2.9871972592708627 min=0.7491972593844356 ls=[-2, -1, 0]
a=-2.236999999886427 min=-2.986130115827886 min=0.7491301159414592 ls=[-2, -1, 0]
a=-2.235999999886427 min=-2.985062934008226 min=0.7490629341217989 ls=[-2, -1, 0]
a=-2.234999999886427 min=-2.9839957137768964 min=0.7489957138904693 ls=[-2, -1, 0]
a=-2.2339999998864273 min=-2.982928455098868 min=0.7489284552124407 ls=[-2, -1, 0]
a=-2.2329999998864274 min=-2.981861157939065 min=0.7488611580526376 ls=[-2, -1, 0]
a=-2.2319999998864275 min=-2.9807938222623678 min=0.7487938223759403 ls=[-2, -1, 0]
a=-2.2309999998864276 min=-2.9797264480336105 min=0.7487264481471831 ls=[-2, -1, 0]
a=-2.2299999998864277 min=-2.978659035217584 min=0.748659035331156 ls=[-2, -1, 0]
a=-2.228999999886428 min=-2.9775915837790308 min=0.7485915838926027 ls=[-2, -1, 0]
a=-2.227999999886428 min=-2.9765240936826505 min=0.7485240937962223 ls=[-2, -1, 0]
a=-2.226999999886428 min=-2.975456564893096 min=0.7484565650066684 ls=[-2, -1, 0]
a=-2.225999999886428 min=-2.9743889973749766 min=0.7483889974885487 ls=[-2, -1, 0]
a=-2.2249999998864283 min=-2.9733213910928535 min=0.7483213912064253 ls=[-2, -1, 0]
a=-2.2239999998864284 min=-2.9722537460112437 min=0.7482537461248153 ls=[-2, -1, 0]
a=-2.2229999998864285 min=-2.9711860620946178 min=0.7481860622081893 ls=[-2, -1, 0]
a=-2.2219999998864286 min=-2.970118339307401 min=0.7481183394209723 ls=[-2, -1, 0]
a=-2.2209999998864287 min=-2.969050577613972 min=0.7480505777275435 ls=[-2, -1, 0]
a=-2.219999999886429 min=-2.9679827769786646 min=0.747982777092236 ls=[-2, -1, 0]
a=-2.218999999886429 min=-2.9669149373657664 min=0.7479149374793372 ls=[-2, -1, 0]
a=-2.217999999886429 min=-2.965847058739517 min=0.7478470588530879 ls=[-2, -1, 0]
a=-2.216999999886429 min=-2.9647791410641116 min=0.7477791411776826 ls=[-2, -1, 0]
a=-2.2159999998864293 min=-2.9637111843036994 min=0.7477111844172701 ls=[-2, -1, 0]
a=-2.2149999998864294 min=-2.9626431884223816 min=0.7476431885359522 ls=[-2, -1, 0]
a=-2.2139999998864295 min=-2.9615751533842145 min=0.7475751534977848 ls=[-2, -1, 0]
a=-2.2129999998864296 min=-2.9605070791532064 min=0.7475070792667768 ls=[-2, -1, 0]
a=-2.2119999998864297 min=-2.9594389656933204 min=0.7474389658068907 ls=[-2, -1, 0]
a=-2.21099999988643 min=-2.9583708129684725 min=0.7473708130820425 ls=[-2, -1, 0]
a=-2.20999999988643 min=-2.9573026209425306 min=0.7473026210561009 ls=[-2, -1, 0]
a=-2.20899999988643 min=-2.9562343895793184 min=0.7472343896928884 ls=[-2, -1, 0]
a=-2.20799999988643 min=-2.95516611884261 min=0.74716611895618 ls=[-2, -1, 0]
a=-2.2069999998864303 min=-2.954097808696134 min=0.747097808809704 ls=[-2, -1, 0]
a=-2.2059999998864304 min=-2.953029459103572 min=0.7470294592171418 ls=[-2, -1, 0]
a=-2.2049999998864305 min=-2.951961070028558 min=0.7469610701421272 ls=[-2, -1, 0]
a=-2.2039999998864306 min=-2.9508926414346774 min=0.7468926415482471 ls=[-2, -1, 0]
a=-2.2029999998864307 min=-2.9498241732854713 min=0.7468241733990406 ls=[-2, -1, 0]
a=-2.201999999886431 min=-2.948755665544431 min=0.7467556656580001 ls=[-2, -1, 0]
a=-2.200999999886431 min=-2.9476871181750006 min=0.7466871182885699 ls=[-2, -1, 0]
a=-2.199999999886431 min=-2.946618531140578 min=0.7466185312541471 ls=[-2, -1, 0]
a=-2.198999999886431 min=-2.945549904404512 min=0.746549904518081 ls=[-2, -1, 0]
a=-2.1979999998864312 min=-2.9444812379301046 min=0.7464812380436732 ls=[-2, -1, 0]
a=-2.1969999998864314 min=-2.9434125316806083 min=0.746412531794177 ls=[-2, -1, 0]
a=-2.1959999998864315 min=-2.9423437856192303 min=0.746343785732799 ls=[-2, -1, 0]
a=-2.1949999998864316 min=-2.9412749997091283 min=0.7462749998226967 ls=[-2, -1, 0]
a=-2.1939999998864317 min=-2.9402061739134115 min=0.7462061740269799 ls=[-2, -1, 0]
a=-2.192999999886432 min=-2.939137308195142 min=0.7461373083087104 ls=[-2, -1, 0]
a=-2.191999999886432 min=-2.9380684025173336 min=0.7460684026309017 ls=[-2, -1, 0]
a=-2.190999999886432 min=-2.9369994568429507 min=0.7459994569565187 ls=[-2, -1, 0]
a=-2.189999999886432 min=-2.9359304711349106 min=0.7459304712484787 ls=[-2, -1, 0]
a=-2.1889999998864322 min=-2.9348614453560815 min=0.7458614454696495 ls=[-2, -1, 0]
a=-2.1879999998864323 min=-2.9337923794692835 min=0.7457923795828512 ls=[-2, -1, 0]
a=-2.1869999998864325 min=-2.9327232734372872 min=0.7457232735508548 ls=[-2, -1, 0]
a=-2.1859999998864326 min=-2.9316541272228154 min=0.7456541273363826 ls=[-2, -1, 0]
a=-2.1849999998864327 min=-2.930584940788541 min=0.7455849409021083 ls=[-2, -1, 0]
a=-2.183999999886433 min=-2.9295157140970893 min=0.7455157142106563 ls=[-2, -1, 0]
a=-2.182999999886433 min=-2.928446447111036 min=0.745446447224603 ls=[-2, -1, 0]
a=-2.181999999886433 min=-2.9273771397929074 min=0.7453771399064744 ls=[-2, -1, 0]
a=-2.180999999886433 min=-2.9263077921051814 min=0.7453077922187483 ls=[-2, -1, 0]
a=-2.1799999998864332 min=-2.925238404010287 min=0.7452384041238533 ls=[-2, -1, 0]
a=-2.1789999998864333 min=-2.924168975470601 min=0.7451689755841677 ls=[-2, -1, 0]
a=-2.1779999998864334 min=-2.9230995064484553 min=0.7450995065620218 ls=[-2, -1, 0]
a=-2.1769999998864336 min=-2.9220299969061285 min=0.7450299970196952 ls=[-2, -1, 0]
a=-2.1759999998864337 min=-2.9209604468058528 min=0.7449604469194189 ls=[-2, -1, 0]
a=-2.1749999998864338 min=-2.9198908561098076 min=0.7448908562233736 ls=[-2, -1, 0]
a=-2.173999999886434 min=-2.9188212247801246 min=0.7448212248936905 ls=[-2, -1, 0]
a=-2.172999999886434 min=-2.917751552778885 min=0.7447515528924509 ls=[-2, -1, 0]
a=-2.171999999886434 min=-2.9166818400681205 min=0.7446818401816861 ls=[-2, -1, 0]
a=-2.170999999886434 min=-2.915612086609812 min=0.7446120867233779 ls=[-2, -1, 0]
a=-2.1699999998864343 min=-2.914542292365892 min=0.7445422924794576 ls=[-2, -1, 0]
a=-2.1689999998864344 min=-2.9134724572982407 min=0.744472457411806 ls=[-2, -1, 0]
a=-2.1679999998864345 min=-2.9124025813686893 min=0.7444025814822546 ls=[-2, -1, 0]
a=-2.1669999998864347 min=-2.9113326645390183 min=0.7443326646525839 ls=[-2, -1, 0]
a=-2.1659999998864348 min=-2.9102627067709586 min=0.7442627068845238 ls=[-2, -1, 0]
a=-2.164999999886435 min=-2.909192708026189 min=0.7441927081397541 ls=[-2, -1, 0]
a=-2.163999999886435 min=-2.908122668266339 min=0.744122668379904 ls=[-2, -1, 0]
a=-2.162999999886435 min=-2.907052587452987 min=0.744052587566552 ls=[-2, -1, 0]
a=-2.161999999886435 min=-2.9059824655476607 min=0.7439824656612253 ls=[-2, -1, 0]
a=-2.1609999998864353 min=-2.9049123025118364 min=0.7439123026254009 ls=[-2, -1, 0]
a=-2.1599999998864354 min=-2.90384209830694 min=0.7438420984205045 ls=[-2, -1, 0]
a=-2.1589999998864355 min=-2.9027718528943467 min=0.7437718530079109 ls=[-2, -1, 0]
a=-2.1579999998864356 min=-2.9017015662353796 min=0.7437015663489439 ls=[-2, -1, 0]
a=-2.1569999998864358 min=-2.9006312382913118 min=0.7436312384048758 ls=[-2, -1, 0]
a=-2.155999999886436 min=-2.8995608690233627 min=0.7435608691369271 ls=[-2, -1, 0]
a=-2.154999999886436 min=-2.898490458392704 min=0.7434904585062683 ls=[-2, -1, 0]
a=-2.153999999886436 min=-2.897420006360454 min=0.7434200064740175 ls=[-2, -1, 0]
a=-2.152999999886436 min=-2.896349512887677 min=0.7433495130012409 ls=[-2, -1, 0]
a=-2.1519999998864363 min=-2.89527897793539 min=0.7432789780489537 ls=[-2, -1, 0]
a=-2.1509999998864364 min=-2.8942084014645557 min=0.7432084015781193 ls=[-2, -1, 0]
a=-2.1499999998864365 min=-2.893137783436085 min=0.7431377835496489 ls=[-2, -1, 0]
a=-2.1489999998864366 min=-2.892067123810839 min=0.743067123924402 ls=[-2, -1, 0]
a=-2.1479999998864368 min=-2.8909964225496223 min=0.7429964226631856 ls=[-2, -1, 0]
a=-2.146999999886437 min=-2.8899256796131922 min=0.7429256797267556 ls=[-2, -1, 0]
a=-2.145999999886437 min=-2.8888548949622517 min=0.7428548950758147 ls=[-2, -1, 0]
a=-2.144999999886437 min=-2.887784068557451 min=0.7427840686710139 ls=[-2, -1, 0]
a=-2.143999999886437 min=-2.8867132003593885 min=0.7427132004729513 ls=[-2, -1, 0]
a=-2.1429999998864373 min=-2.8856422903286103 min=0.742642290442173 ls=[-2, -1, 0]
a=-2.1419999998864374 min=-2.8845713384256095 min=0.7425713385391723 ls=[-2, -1, 0]
a=-2.1409999998864375 min=-2.8835003446108276 min=0.7425003447243903 ls=[-2, -1, 0]
a=-2.1399999998864376 min=-2.8824293088446518 min=0.7424293089582141 ls=[-2, -1, 0]
a=-2.1389999998864377 min=-2.8813582310874173 min=0.7423582312009793 ls=[-2, -1, 0]
a=-2.137999999886438 min=-2.880287111299406 min=0.7422871114129681 ls=[-2, -1, 0]
a=-2.136999999886438 min=-2.879215949440847 min=0.7422159495544092 ls=[-2, -1, 0]
a=-2.135999999886438 min=-2.878144745471917 min=0.7421447455854788 ls=[-2, -1, 0]
a=-2.134999999886438 min=-2.8770734993527376 min=0.7420734994662994 ls=[-2, -1, 0]
a=-2.1339999998864383 min=-2.876002211043379 min=0.7420022111569409 ls=[-2, -1, 0]
a=-2.1329999998864384 min=-2.8749308805038574 min=0.741930880617419 ls=[-2, -1, 0]
a=-2.1319999998864385 min=-2.8738595076941347 min=0.7418595078076962 ls=[-2, -1, 0]
a=-2.1309999998864386 min=-2.87278809257412 min=0.7417880926876814 ls=[-2, -1, 0]
a=-2.1299999998864387 min=-2.8717166351036685 min=0.7417166352172297 ls=[-2, -1, 0]
a=-2.128999999886439 min=-2.870645135242582 min=0.7416451353561433 ls=[-2, -1, 0]
a=-2.127999999886439 min=-2.8695735929506077 min=0.7415735930641689 ls=[-2, -1, 0]
a=-2.126999999886439 min=-2.8685020081874395 min=0.7415020083010004 ls=[-2, -1, 0]
a=-2.125999999886439 min=-2.867430380912716 min=0.7414303810262772 ls=[-2, -1, 0]
a=-2.1249999998864393 min=-2.8663587110860247 min=0.7413587111995852 ls=[-2, -1, 0]
a=-2.1239999998864394 min=-2.8652869986668943 min=0.7412869987804549 ls=[-2, -1, 0]
a=-2.1229999998864395 min=-2.8642152436148027 min=0.7412152437283632 ls=[-2, -1, 0]
a=-2.1219999998864396 min=-2.8631434458891722 min=0.7411434460027326 ls=[-2, -1, 0]
a=-2.1209999998864397 min=-2.8620716054493704 min=0.7410716055629307 ls=[-2, -1, 0]
a=-2.11999999988644 min=-2.8609997222547108 min=0.7409997223682707 ls=[-2, -1, 0]
a=-2.11899999988644 min=-2.8599277962644507 min=0.7409277963780108 ls=[-2, -1, 0]
a=-2.11799999988644 min=-2.858855827437795 min=0.7408558275513546 ls=[-2, -1, 0]
a=-2.11699999988644 min=-2.857783815733891 min=0.7407838158474509 ls=[-2, -1, 0]
a=-2.1159999998864403 min=-2.8567117611118333 min=0.7407117612253931 ls=[-2, -1, 0]
a=-2.1149999998864404 min=-2.85563966353066 min=0.7406396636442196 ls=[-2, -1, 0]
a=-2.1139999998864405 min=-2.8545675229493543 min=0.7405675230629138 ls=[-2, -1, 0]
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a=6.102000000113811 min=1.2603015329124712 min=4.84169846720134 ls=[2, 3, 4]
a=6.103000000113811 min=1.260228873110043 min=4.842771127003768 ls=[2, 3, 4]
a=6.104000000113811 min=1.2601562568042914 min=4.84384374330952 ls=[2, 3, 4]
a=6.105000000113812 min=1.2600836839535297 min=4.8449163161602815 ls=[2, 3, 4]
a=6.106000000113812 min=1.2600111545161268 min=4.845988845597685 ls=[2, 3, 4]
a=6.107000000113812 min=1.2599386684505087 min=4.847061331663303 ls=[2, 3, 4]
a=6.108000000113813 min=1.259866225715157 min=4.848133774398656 ls=[2, 3, 4]
a=6.109000000113813 min=1.2597938262686093 min=4.849206173845204 ls=[2, 3, 4]
a=6.110000000113813 min=1.2597214700694603 min=4.850278530044353 ls=[2, 3, 4]
a=6.111000000113814 min=1.259649157076359 min=4.851350843037455 ls=[2, 3, 4]
a=6.112000000113814 min=1.2595768872480113 min=4.852423112865803 ls=[2, 3, 4]
a=6.113000000113814 min=1.2595046605431792 min=4.853495339570635 ls=[2, 3, 4]
a=6.114000000113815 min=1.2594324769206786 min=4.854567523193136 ls=[2, 3, 4]
a=6.115000000113815 min=1.2593603363393826 min=4.855639663774433 ls=[2, 3, 4]
a=6.116000000113815 min=1.2592882387582187 min=4.856711761355596 ls=[2, 3, 4]
a=6.117000000113816 min=1.2592161841361706 min=4.857783815977645 ls=[2, 3, 4]
a=6.118000000113816 min=1.2591441724322767 min=4.8588558276815395 ls=[2, 3, 4]
a=6.119000000113816 min=1.25907220360563 min=4.859927796508186 ls=[2, 3, 4]
a=6.120000000113817 min=1.2590002776153797 min=4.860999722498437 ls=[2, 3, 4]
a=6.121000000113817 min=1.2589283944207295 min=4.862071605693087 ls=[2, 3, 4]
a=6.122000000113817 min=1.258856553980937 min=4.86314344613288 ls=[2, 3, 4]
a=6.123000000113818 min=1.2587847562553165 min=4.864215243858501 ls=[2, 3, 4]
a=6.124000000113818 min=1.2587130012032341 min=4.865286998910584 ls=[2, 3, 4]
a=6.125000000113818 min=1.2586412887841136 min=4.866358711329704 ls=[2, 3, 4]
a=6.126000000113819 min=1.2585696189574314 min=4.867430381156387 ls=[2, 3, 4]
a=6.127000000113819 min=1.2584979916827177 min=4.8685020084311015 ls=[2, 3, 4]
a=6.128000000113819 min=1.258426406919559 min=4.869573593194261 ls=[2, 3, 4]
a=6.12900000011382 min=1.2583548646275942 min=4.8706451354862255 ls=[2, 3, 4]
a=6.13000000011382 min=1.2582833647665175 min=4.871716635347303 ls=[2, 3, 4]
a=6.13100000011382 min=1.2582119072960751 min=4.872788092817745 ls=[2, 3, 4]
a=6.132000000113821 min=1.2581404921760702 min=4.87385950793775 ls=[2, 3, 4]
a=6.133000000113821 min=1.2580691193663571 min=4.874930880747463 ls=[2, 3, 4]
a=6.134000000113821 min=1.2579977888268445 min=4.876002211286977 ls=[2, 3, 4]
a=6.135000000113822 min=1.2579265005174955 min=4.877073499596326 ls=[2, 3, 4]
a=6.136000000113822 min=1.2578552543983257 min=4.878144745715496 ls=[2, 3, 4]
a=6.137000000113822 min=1.257784050429405 min=4.879215949684417 ls=[2, 3, 4]
a=6.138000000113823 min=1.2577128885708555 min=4.880287111542967 ls=[2, 3, 4]
a=6.139000000113823 min=1.2576417687828536 min=4.881358231330969 ls=[2, 3, 4]
a=6.140000000113823 min=1.2575706910256288 min=4.882429309088194 ls=[2, 3, 4]
a=6.141000000113824 min=1.257499655259462 min=4.883500344854362 ls=[2, 3, 4]
a=6.142000000113824 min=1.2574286614446892 min=4.884571338669135 ls=[2, 3, 4]
a=6.143000000113824 min=1.257357709541698 min=4.885642290572126 ls=[2, 3, 4]
a=6.144000000113825 min=1.2572867995109296 min=4.886713200602895 ls=[2, 3, 4]
a=6.145000000113825 min=1.2572159313128766 min=4.887784068800949 ls=[2, 3, 4]
a=6.146000000113825 min=1.2571451049080853 min=4.88885489520574 ls=[2, 3, 4]
a=6.147000000113826 min=1.2570743202571535 min=4.889925679856672 ls=[2, 3, 4]
a=6.148000000113826 min=1.257003577320733 min=4.890996422793092 ls=[2, 3, 4]
a=6.149000000113826 min=1.2569328760595264 min=4.8920671240543 ls=[2, 3, 4]
a=6.150000000113827 min=1.2568622164342886 min=4.893137783679538 ls=[2, 3, 4]
a=6.151000000113827 min=1.2567915984058273 min=4.894208401707999 ls=[2, 3, 4]
a=6.152000000113827 min=1.2567210219350025 min=4.895278978178824 ls=[2, 3, 4]
a=6.153000000113828 min=1.2566504869827246 min=4.896349513131103 ls=[2, 3, 4]
a=6.154000000113828 min=1.2565799935099573 min=4.897420006603871 ls=[2, 3, 4]
a=6.155000000113828 min=1.256509541477716 min=4.898490458636112 ls=[2, 3, 4]
a=6.156000000113829 min=1.2564391308470666 min=4.899560869266762 ls=[2, 3, 4]
a=6.157000000113829 min=1.2563687615791275 min=4.900631238534702 ls=[2, 3, 4]
a=6.158000000113829 min=1.2562984336350687 min=4.901701566478761 ls=[2, 3, 4]
a=6.15900000011383 min=1.2562281469761112 min=4.902771853137718 ls=[2, 3, 4]
a=6.16000000011383 min=1.256157901563527 min=4.903842098550303 ls=[2, 3, 4]
a=6.16100000011383 min=1.2560876973586397 min=4.904912302755191 ls=[2, 3, 4]
a=6.162000000113831 min=1.2560175343228248 min=4.905982465791006 ls=[2, 3, 4]
a=6.163000000113831 min=1.2559474124175074 min=4.907052587696324 ls=[2, 3, 4]
a=6.164000000113831 min=1.2558773316041645 min=4.908122668509667 ls=[2, 3, 4]
a=6.165000000113832 min=1.2558072918443235 min=4.909192708269508 ls=[2, 3, 4]
a=6.166000000113832 min=1.2557372930995632 min=4.910262707014269 ls=[2, 3, 4]
a=6.167000000113832 min=1.2556673353315124 min=4.91133266478232 ls=[2, 3, 4]
a=6.168000000113833 min=1.255597418501851 min=4.912402581611982 ls=[2, 3, 4]
a=6.169000000113833 min=1.2555275425723091 min=4.913472457541523 ls=[2, 3, 4]
a=6.170000000113833 min=1.255457707504667 min=4.914542292609166 ls=[2, 3, 4]
a=6.171000000113834 min=1.2553879132607557 min=4.915612086853078 ls=[2, 3, 4]
a=6.172000000113834 min=1.2553181598024563 min=4.9166818403113774 ls=[2, 3, 4]
a=6.173000000113834 min=1.2552484470917011 min=4.917751553022133 ls=[2, 3, 4]
a=6.174000000113835 min=1.2551787750904706 min=4.918821225023364 ls=[2, 3, 4]
a=6.175000000113835 min=1.2551091437607969 min=4.919890856353038 ls=[2, 3, 4]
a=6.176000000113835 min=1.255039553064761 min=4.920960447049074 ls=[2, 3, 4]
a=6.177000000113836 min=1.2549700029644937 min=4.922029997149342 ls=[2, 3, 4]
a=6.178000000113836 min=1.2549004934221764 min=4.92309950669166 ls=[2, 3, 4]
a=6.179000000113836 min=1.2548310244000391 min=4.924168975713798 ls=[2, 3, 4]
a=6.180000000113837 min=1.254761595860363 min=4.925238404253474 ls=[2, 3, 4]
a=6.181000000113837 min=1.2546922077654767 min=4.926307792348361 ls=[2, 3, 4]
a=6.182000000113837 min=1.2546228600777602 min=4.927377140036077 ls=[2, 3, 4]
a=6.183000000113838 min=1.254553552759641 min=4.9284464473541965 ls=[2, 3, 4]
a=6.184000000113838 min=1.2544842857735965 min=4.9295157143402415 ls=[2, 3, 4]
a=6.185000000113838 min=1.254415059082154 min=4.930584941031684 ls=[2, 3, 4]
a=6.186000000113839 min=1.2543458726478887 min=4.93165412746595 ls=[2, 3, 4]
a=6.187000000113839 min=1.2542767264334258 min=4.932723273680413 ls=[2, 3, 4]
a=6.188000000113839 min=1.2542076204014383 min=4.933792379712401 ls=[2, 3, 4]
a=6.18900000011384 min=1.2541385545146488 min=4.934861445599191 ls=[2, 3, 4]
a=6.19000000011384 min=1.2540695287358288 min=4.935930471378011 ls=[2, 3, 4]
a=6.19100000011384 min=1.2540005430277978 min=4.936999457086042 ls=[2, 3, 4]
a=6.192000000113841 min=1.253931597353424 min=4.9380684027604165 ls=[2, 3, 4]
a=6.193000000113841 min=1.2538626916756244 min=4.939137308438217 ls=[2, 3, 4]
a=6.194000000113841 min=1.253793825957364 min=4.940206174156478 ls=[2, 3, 4]
a=6.195000000113842 min=1.2537250001616562 min=4.941274999952185 ls=[2, 3, 4]
a=6.196000000113842 min=1.253656214251563 min=4.942343785862279 ls=[2, 3, 4]
a=6.197000000113842 min=1.2535874681901937 min=4.943412531923649 ls=[2, 3, 4]
a=6.198000000113843 min=1.253518761940707 min=4.944481238173136 ls=[2, 3, 4]
a=6.199000000113843 min=1.253450095466308 min=4.945549904647535 ls=[2, 3, 4]
a=6.200000000113843 min=1.253381468730251 min=4.946618531383592 ls=[2, 3, 4]
a=6.201000000113844 min=1.2533128816958372 min=4.947687118418006 ls=[2, 3, 4]
a=6.202000000113844 min=1.253244334326416 min=4.948755665787428 ls=[2, 3, 4]
a=6.203000000113844 min=1.253175826585384 min=4.9498241735284605 ls=[2, 3, 4]
a=6.204000000113845 min=1.2531073584361867 min=4.950892641677658 ls=[2, 3, 4]
a=6.205000000113845 min=1.2530389298423155 min=4.95196107027153 ls=[2, 3, 4]
a=6.206000000113845 min=1.2529705407673102 min=4.953029459346535 ls=[2, 3, 4]
a=6.207000000113846 min=1.2529021911747564 min=4.954097808939089 ls=[2, 3, 4]
a=6.208000000113846 min=1.2528338810282895 min=4.955166119085557 ls=[2, 3, 4]
a=6.209000000113846 min=1.2527656102915905 min=4.956234389822256 ls=[2, 3, 4]
a=6.210000000113847 min=1.2526973789283866 min=4.95730262118546 ls=[2, 3, 4]
a=6.211000000113847 min=1.2526291869024542 min=4.958370813211393 ls=[2, 3, 4]
a=6.212000000113847 min=1.2525610341776146 min=4.959438965936233 ls=[2, 3, 4]
a=6.213000000113848 min=1.2524929207177373 min=4.960507079396111 ls=[2, 3, 4]
a=6.214000000113848 min=1.2524248464867382 min=4.96157515362711 ls=[2, 3, 4]
a=6.215000000113848 min=1.2523568114485797 min=4.962643188665268 ls=[2, 3, 4]
a=6.216000000113849 min=1.2522888155672705 min=4.963711184546578 ls=[2, 3, 4]
a=6.217000000113849 min=1.252220858806867 min=4.964779141306982 ls=[2, 3, 4]
a=6.218000000113849 min=1.2521529411314707 min=4.965847058982378 ls=[2, 3, 4]
a=6.21900000011385 min=1.2520850625052302 min=4.966914937608619 ls=[2, 3, 4]
a=6.22000000011385 min=1.2520172228923403 min=4.9679827772215095 ls=[2, 3, 4]
a=6.22100000011385 min=1.2519494222570415 min=4.969050577856809 ls=[2, 3, 4]
a=6.222000000113851 min=1.2518816605636216 min=4.970118339550229 ls=[2, 3, 4]
a=6.223000000113851 min=1.2518139377764137 min=4.971186062337438 ls=[2, 3, 4]
a=6.224000000113851 min=1.251746253859796 min=4.972253746254055 ls=[2, 3, 4]
a=6.225000000113852 min=1.2516786087781948 min=4.973321391335657 ls=[2, 3, 4]
a=6.226000000113852 min=1.2516110024960805 min=4.974388997617772 ls=[2, 3, 4]
a=6.227000000113852 min=1.2515434349779697 min=4.975456565135882 ls=[2, 3, 4]
a=6.228000000113853 min=1.251475906188424 min=4.976524093925429 ls=[2, 3, 4]
a=6.229000000113853 min=1.2514084160920529 min=4.9775915840218 ls=[2, 3, 4]
a=6.230000000113853 min=1.2513409646535083 min=4.978659035460345 ls=[2, 3, 4]
a=6.231000000113854 min=1.2512735518374896 min=4.979726448276364 ls=[2, 3, 4]
a=6.232000000113854 min=1.2512061776087413 min=4.980793822505113 ls=[2, 3, 4]
a=6.233000000113854 min=1.251138841932053 min=4.981861158181801 ls=[2, 3, 4]
a=6.234000000113855 min=1.2510715447722585 min=4.982928455341597 ls=[2, 3, 4]
a=6.235000000113855 min=1.2510042860942385 min=4.983995714019617 ls=[2, 3, 4]
a=6.236000000113855 min=1.2509370658629178 min=4.985062934250937 ls=[2, 3, 4]
a=6.237000000113856 min=1.250869884043266 min=4.98613011607059 ls=[2, 3, 4]
a=6.238000000113856 min=1.2508027406002986 min=4.987197259513557 ls=[2, 3, 4]
a=6.239000000113856 min=1.2507356354990746 min=4.988264364614782 ls=[2, 3, 4]
a=6.240000000113857 min=1.2506685687046986 min=4.989331431409158 ls=[2, 3, 4]
a=6.241000000113857 min=1.2506015401823198 min=4.990398459931537 ls=[2, 3, 4]
a=6.242000000113857 min=1.2505345498971316 min=4.991465450216726 ls=[2, 3, 4]
a=6.243000000113858 min=1.2504675978143733 min=4.992532402299484 ls=[2, 3, 4]
a=6.244000000113858 min=1.2504006838993262 min=4.993599316214532 ls=[2, 3, 4]
a=6.245000000113858 min=1.2503338081173183 min=4.99466619199654 ls=[2, 3, 4]
a=6.246000000113859 min=1.2502669704337208 min=4.995733029680138 ls=[2, 3, 4]
a=6.247000000113859 min=1.2502001708139492 min=4.99679982929991 ls=[2, 3, 4]
a=6.248000000113859 min=1.2501334092234633 min=4.997866590890396 ls=[2, 3, 4]
a=6.24900000011386 min=1.2500666856277673 min=4.998933314486092 ls=[2, 3, 4]

从以上结果来看，a∈（-2.25，-1.33）时，两根间的整数是[-2, -1, 0]；

a∈（5.33，6.25时），两根间的整数是[2, 3, 4]；

程序结果和传统解法是一致的。

【函数图像】

当a取6时，函数图像为

当取-2.5时，函数图像为

由图可见，两根间的整数确实符合题目和解答。

【勾画图线的Canvas代码】

<!DOCTYPE html>
<html lang="utf-8">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<head>
     <title>UNASSIGNED</title>
     <style type="text/css">
     .centerlize{
        margin:0 auto;
        border:0px solid red;
        width:1200px;height:600px;
     }
     </style>
     </head>

     <body onload="draw();">
        <div class="centerlize">
            <canvas id="myCanvas" width="1200px" height="600px" style="border:1px dashed black;">
                如果看到这段文字说您的浏览器尚不支持HTML5 Canvas，请更换浏览器再试.
            </canvas>
        </div>
     </body>
</html>
<script type="text/javascript">
<!--
/*****************************************************************
* 将全体代码(从<!DOCTYPE到script>)拷贝下来，粘贴到文本编辑器中，
* 另存为.html文件，再用chrome浏览器打开，就能看到实现效果。
******************************************************************/
// 画布宽度
const WIDTH=1200;

// 画布高度
const HEIGHT=600;

// 画布环境
var context=0;   

// 缩放比例
const ScaleUnit=50;

// 舞台对象
var stage;

// 消逝的时间
var timeElapsed=0;

// 图像标题，title
const TITLE=" y=x^2+2.5x-2.5";

// 核心勾画函数，由body_onload调用
function draw(){
    document.title="函数"+TITLE+" 图示";

    // 画图前初始化
    var canvas=document.getElementById('myCanvas');    
    canvas.width=WIDTH;
    canvas.height=HEIGHT; 
    context=canvas.getContext('2d');  
    
    // 进行屏幕坐标系到笛卡尔坐标系的变换
    // 处置完成前，原点在左上角，向右为X正向，向下为Y的正向
    // 处置完毕后，原点移动到画布中央，向右为X正向，向上为Y的正向
    context.translate(WIDTH/2,HEIGHT/2);
    context.rotate(getRad(180));
    context.scale(-1,1);

    //context.translate(0,0);
    //context.scale(1,1);

    // 初始化舞台
    stage=new Stage();

    // 开始动画
    animate();
};

//-------------------------------
// 画图
//-------------------------------
function animate(){    
    timeElapsed+=1;// 时间每轮增加1

    stage.update(timeElapsed);
    stage.paintBg(context);
    stage.paint(context);

    if(timeElapsed<2400){        
        window.requestAnimationFrame(animate);
    }
}

//-------------------------------
// 舞台对象定义处
//-------------------------------
function Stage(){
    var obj=new Object;

    obj.prpty={"x":-12,"y":0,"pts":[],"pts2":[],"max":-10,"min":10};

    // 随时间更新位置
    obj.update=function(t){
        // xy值是如何变化的
        obj.prpty.x+=0.01;
        let x=obj.prpty.x;
        obj.prpty.y=x*x+2.5*x-2.5;

        // 取极值
        if(obj.prpty.y<obj.prpty.min){
            obj.prpty.min=obj.prpty.y;
        }
        if(obj.prpty.y>obj.prpty.max){
            obj.prpty.max=obj.prpty.y;
        }

        // 放入数组
        let arr={"x":obj.prpty.x,"y":obj.prpty.y};

        //if(arr.x<5){
            this.prpty.pts.push(arr);
        //}else{
        //    this.prpty.pts2.push(arr);
        //}
        
    };

    // 画前景
    obj.paint=function(ctx){  
        // 写当前点坐标
        drawText(ctx,"当前 X:"+this.prpty.x.toFixed(3)+"  Y:"+this.prpty.y.toFixed(3),-400,125,"navy",18);

        // 写极值
        drawText(ctx,"max:"+this.prpty.max.toFixed(3)+"  min:"+this.prpty.min.toFixed(3),-400,105,"navy",18);

        // 绘制曲线
        paintCurve(ctx,"maroon",this.prpty.pts);
        //paintCurve(ctx,"orange",this.prpty.pts2);
    };

    // 画背景
    obj.paintBg=function(ctx){
        // 清屏
        ctx.clearRect(-600,-300,1200,600);

        // 画X轴
        drawAxisX(ctx,-600,600,50);

        // 画Y轴
        drawAxisY(ctx,-300,300,50);

        // 画网格线
        drawGrid(ctx,-600,-300,50,1200,600,50,"grey");
        
        // 标题
        drawText(ctx,"函数 "+TITLE+" 图示",-400,-160,"navy",18);

        // 作者，日期
        drawText(ctx,"逆火",-500,-200,"navy",16);
        drawText(ctx,(new Date()).toLocaleDateString(),-500,-220,"navy",16);
    };
    
    return obj;
}

// 连点成线画曲线
function paintCurve(ctx,color,cds){
    ctx.strokeStyle = color;
    ctx.beginPath(); 
    
    for(var i=0; i<cds.length; i++){
        let y=cds[i].y;
        if(Math.abs(cds[i].y*ScaleUnit)<300){
            ctx.lineTo(cds[i].x*ScaleUnit,cds[i].y*ScaleUnit);
        }
    }     
    
    ctx.stroke();
    ctx.closePath();
}

// 定点画实心圆
function drawSolidCircle(ctx,x,y,r,color){
      ctx.save();

      ctx.beginPath();
      ctx.arc(x,y,r,0,2*Math.PI);
      ctx.fillStyle=color;
      ctx.fill();
      ctx.stroke();
      ctx.restore();
}

// 两点之间画线段
function drawLine(ctx,x1,y1,x2,y2,color){
    ctx.save();

    ctx.lineWidth=0.25;
    ctx.strokeStyle=color;
    ctx.fillStyle=color;

    ctx.beginPath();
    ctx.moveTo(x1,y1);
    ctx.lineTo(x2,y2);
    ctx.stroke();
    ctx.closePath();

    ctx.restore();
}

// 画横轴
function drawAxisX(ctx,start,end,step){
    ctx.save();
    
    ctx.lineWidth=0.25;
    ctx.strokeStyle='navy';
    ctx.fillStyle='navy';

    // 画轴
    ctx.beginPath();
    ctx.moveTo(start, 0);
    ctx.lineTo(end, 0);
    ctx.stroke();
    ctx.closePath();

    // 画箭头
    ctx.beginPath();
    ctx.moveTo(end-Math.cos(getRad(15))*10, Math.sin(getRad(15))*10);
    ctx.lineTo(end, 0);
    ctx.lineTo(end-Math.cos(getRad(15))*10, -Math.sin(getRad(15))*10);
    ctx.stroke();
    ctx.closePath();
    
    // 画刻度
    var x,y;
    y=5;
    for(x=start;x<end;x+=step){
        ctx.beginPath();
        ctx.moveTo(x, 0);
        ctx.lineTo(x, y);
        
        ctx.stroke();
        ctx.closePath();

        drawText(ctx,x/ScaleUnit+"",x,y-20,"navy",12);
    }

    ctx.restore();
}

// 画纵轴
function drawAxisY(ctx,start,end,step){
    ctx.save();
    
    ctx.lineWidth=0.5;
    ctx.strokeStyle='navy';
    ctx.fillStyle='navy';

    // 画轴
    ctx.beginPath();
    ctx.moveTo(0, start);
    ctx.lineTo(0, end);
    ctx.stroke();
    ctx.closePath();

    // 画箭头
    ctx.beginPath();
    ctx.moveTo(Math.sin(getRad(15))*10, end-Math.cos(getRad(15))*10);
    ctx.lineTo(0, end);
    ctx.lineTo(-Math.sin(getRad(15))*10, end-Math.cos(getRad(15))*10);
    ctx.stroke();
    ctx.closePath();
    
    // 画刻度
    var x,y;
    x=5;
    for(y=start;y<end;y+=step){
        ctx.beginPath();
        ctx.moveTo(x, y);
        ctx.lineTo(0, y);
        
        drawText(ctx,y/ScaleUnit+"",x-15,y,"navy",12);

        ctx.stroke();
        ctx.closePath();
    }

    ctx.restore();
}

// 画网格线
function drawGrid(ctx,x1,y1,step1,x2,y2,step2,color){
    ctx.save();
    
    ctx.lineWidth=0.5;
    ctx.strokeStyle=color;
    ctx.fillStyle=color;

    var x,y;
    for(x=x1;x<x2;x+=step1){
        ctx.beginPath();
        ctx.moveTo(x, y1);
        ctx.lineTo(x, y2);
        ctx.stroke();
        ctx.closePath();
    }

    for(y=y1;y<y2;y+=step2){
        ctx.beginPath();
        ctx.moveTo(x1, y);
        ctx.lineTo(x2, y);
        ctx.stroke();
        ctx.closePath();
    }

    ctx.restore();
}

//-------------------------------
// 角度得到弧度
//-------------------------------
function getRad(degree){
    return degree/180*Math.PI;
}

//-------------------------------
// 得到颜色
//-------------------------------
function getColor(index){
    var arr=[
        "aqua"/* aqua湖绿色*/,
        "black"/* black黑色*/,
        "blue"/* blue蓝色*/,
        "fuchsia"/* fuchsia 紫红*/,
        "green"/* green 绿色*/,
        "grey"/* grey 草木灰*/,
        "lime"/* lime 亮绿色*/,
        "maroon"/* maroon 棕色*/,
        "navy"/* navy 海军蓝*/,
        "orange"/* orange 橙色*/,
        "purple"/* purple 紫色*/,
        "red"/* red 大红*/,        
        "skyblue"/* skyblue 天蓝*/,
        "teal"/* teal 蓝绿色*/,
        "yellow"/* yellow 亮黄*/,
        "#aa0000"/* #aa0000 铁锈红*/,        
    ];

    if(index>arr.length){
        index=index % arr.length;
    }

    return arr[index];
}

//-------------------------------------
// 绘制文字,指定颜色
// ctx:绘图环境
// text:文字
// x,y:坐标
// color:颜色
// size:字体大小
//-------------------------------------
function drawText(ctx,text,x,y,color,size){
    ctx.save();
    ctx.translate(x,y)
    ctx.rotate(getRad(180))
    ctx.scale(-1,1)

    ctx.textBaseline="bottom";
    ctx.textAlign="center";
    ctx.fillStyle=color;
    ctx.font = size+"px consolas";

    ctx.fillText(text,0,0);
    ctx.restore();
}
//-->
</script>

【结论】

一道数学题如果存在多种解法，那么最简洁的解法为最佳路径。

从这个意义上来说，程序解法已经比传统解法更优越。

END