一元二次方程應(yīng)用題-

1、一元二次方程應(yīng)用題初三數(shù)學(xué)主講教師：林素卿 前面我們學(xué)習(xí)了一元二次方程及其解法，這一節(jié)我們將學(xué)習(xí)列一元二次方程解應(yīng)用題.分三小節(jié)學(xué)習(xí)：數(shù)字問題、面積問題和增長率問題. 列方程解應(yīng)用題的一般步驟：列方程解應(yīng)用題的一般步驟：（1）仔細(xì)分析題意，適當(dāng)?shù)丶僭O(shè)某個(gè)未知量為未知數(shù)；（2）根據(jù)題目中的等量關(guān)系列出方程；（3）解方程，得到方程的解；（4）檢驗(yàn)方程的解是否符合實(shí)際，得到原問題的解；（5）答題.一一 、數(shù)字問題、數(shù)字問題例例1 已知兩個(gè)數(shù)的差是已知兩個(gè)數(shù)的差是8，積是，積是209，求這兩個(gè)數(shù)，求這兩個(gè)數(shù). 解：設(shè)較小的數(shù)為x，則較大的數(shù)為(x+8)，根據(jù)題意，得x(x8) 209x28x16209

2、+16(x4)2225x415x111, x219當(dāng)x=11時(shí)，x819；當(dāng)x19時(shí)，x811.都符合題意.答：這兩個(gè)數(shù)分別11和19，或19和11. 例例2 三個(gè)連續(xù)偶數(shù)，已知最大數(shù)與最小數(shù)的平方和比三個(gè)連續(xù)偶數(shù)，已知最大數(shù)與最小數(shù)的平方和比中間一個(gè)數(shù)的平方大中間一個(gè)數(shù)的平方大332，求這三個(gè)連續(xù)偶數(shù)，求這三個(gè)連續(xù)偶數(shù). 1、偶數(shù)個(gè)連續(xù)偶數(shù)（或奇數(shù)），一般可設(shè)中間兩個(gè)為(x1)和(x 1).2、奇數(shù)個(gè)連續(xù)偶數(shù)（或奇數(shù)，自然數(shù)），一般可設(shè)中間一個(gè)為x.如三個(gè)連續(xù)偶數(shù)，可設(shè)中間一個(gè)偶數(shù)為x，則其余兩個(gè)偶數(shù)分別為(x2)和(x+2)又如三個(gè)連續(xù)自然數(shù)，可設(shè)中間一個(gè)自然數(shù)為x，則其余兩個(gè)自然數(shù)分別為

3、(x1)和(x 1).解解：設(shè)中間一個(gè)偶數(shù)為x，則其余兩個(gè)偶數(shù)分別為(x2)和 (x2),根據(jù)題意，得(x2)2+(x2)2 x2 332整理，得 x2 324 x18當(dāng)x18時(shí)，x2 16, x2 20；當(dāng)x= 18時(shí)，x2= 20, x2 16.答：這三個(gè)連續(xù)偶數(shù)分別為16、18和20，或20、 18和16. 例例3 一個(gè)兩位數(shù)等于其各位數(shù)字之積的一個(gè)兩位數(shù)等于其各位數(shù)字之積的3倍，且其十倍，且其十位數(shù)字比個(gè)位數(shù)字小位數(shù)字比個(gè)位數(shù)字小2，求這個(gè)兩位數(shù)，求這個(gè)兩位數(shù). 解：設(shè)這個(gè)兩位數(shù)的十位數(shù)字為x，則其個(gè)位數(shù)字為（x2），根據(jù)題意，得10 x(x2)3x(x2)整理，得 3x25x20解得

4、 不合題意，舍去；x2, 10 x(x2)24.答：這個(gè)兩位數(shù)為24. 121,2.3xx 13x （1）十位數(shù)字為a，個(gè)位數(shù)字為b的兩位數(shù)是10ab;（2）百位數(shù)字為a，十位數(shù)字為b，個(gè)位數(shù)字為c的三 位數(shù)是100a10bc. 二、面積問題二、面積問題 例例4 有一塊長4米，寬3米的長方形空地，現(xiàn)要在空地中央建一個(gè)長方形花壇，四周是等寬的草坪，使花壇面積是草坪面積的兩倍，求花壇的長和寬.(精確到0.1米) 解：設(shè)四周的等寬草坪的寬為x米，則花壇的長和寬分別為(42x)米和(32x)米，根據(jù)題意，得 (42x)(32x)243(42x)(32x)整理，得 2x27x20解得 ， .當(dāng)x3.19

5、時(shí)，42x0, 不合題意，舍去；當(dāng)x0.31時(shí)，42x3.383.4, 32x2.382.4. 符合題意.答：花壇的長約為3.4米，寬約為2.4米.17333.194x27330.314x例例5 學(xué)校要建一個(gè)面積為150平方米的長方形自行車棚，為節(jié)約經(jīng)費(fèi)，一邊利用18米長的教學(xué)樓后墻，另三邊利用總長為35米的鐵圍欄圍成，求自行車棚的長和寬.解：解：設(shè)與教學(xué)樓后墻垂直的一條邊長為x米，則與教學(xué)樓后墻平行的那條邊長為(352x)米，根據(jù)題意，得 x(352x)150解得 當(dāng) 時(shí)，352x2018不合題意，舍去；當(dāng)x10時(shí)，352x15. 符合題意.答：自行車棚的長和寬分別為15米和10米.1215

6、,10.2xx152x 例例6 如圖，一塊長162米，寬64米的矩形耕地，要在這塊地上挖4橫2縱共6條水渠，如果水渠的寬相等，且要保證余下的可耕地面積為9600平方米，求水渠的寬.(精確到0.1米)解：解：設(shè)水渠的寬為x米，根據(jù)題意，得 464x2（1624x）x960016264整理，得 2x2145x1920解得 不合題意,舍去； 符合題意.答：水渠的寬約為1.3米.211451454 2 19271.2644x 221451454 2 1921.34x 三、增長率問題三、增長率問題例例7 某市市政府計(jì)劃兩年后實(shí)現(xiàn)市財(cái)政凈收入翻一番，那么這兩年中市財(cái)政凈收入的平均年增長率應(yīng)為多少？(精確到

7、0.1%)若記今年市財(cái)政凈收入為a,市財(cái)政凈收入的平均年增長率為x，則明年（一年后）市財(cái)政凈收入為aaxa(1x);后年（兩年后）市財(cái)政凈收入為a(1x) a(1x) xa(1x)2.解：解：設(shè)今年某市市財(cái)政凈收入為1，這兩年中市財(cái)政凈收入的平均年增長率為x，根據(jù)題意，得 (1x)22 , 符合題意. 不合題意，舍去.答：這兩年中市財(cái)政凈收入的平均年增長率約為41.4%.12x 12x 1120.41441.4%x 2120 x 例例8某服裝店原計(jì)劃按每套200元的價(jià)格銷售一批保暖內(nèi)衣，但上市后銷售不佳，為減少庫存積壓，兩次連續(xù)降價(jià)打折處理，最后價(jià)格調(diào)整為每套128元.若兩次降價(jià)折扣率相同,求

8、每次降價(jià)率為多少？兩次打折標(biāo)示多少折？（1）若原價(jià)為a元，降價(jià)率為x，則一次降價(jià)后標(biāo)價(jià)為 aax a(1x) 元; 兩次降價(jià)后標(biāo)價(jià)為a(1x)2元.（2）若原價(jià)為a元，打y折，則一次打折后標(biāo)價(jià)為 元; 兩次降價(jià)后標(biāo)價(jià)為元. 降價(jià)率x與打y折的關(guān)系為: .110yx10ya2()10ya解：解：設(shè)每次降價(jià)率為x, 根據(jù)題意，得 200(1x)2128 x110.80.220%,符合題意，此時(shí)兩次打折標(biāo)示均為8折； x210.81不合題意，舍去.答：每次降價(jià)率均為20%，兩次打折標(biāo)示均為8折.10.640.8x 例例9 小明將勤工儉學(xué)得到的500元錢按一年定期存入銀行，到期后取出50元用來購買學(xué)習(xí)

9、用品，剩下的450元連同應(yīng)得稅后利息又全部按一年定期存入銀行，到期后可得稅后本息約461元（利息稅為利息的20%），如果這個(gè)期間銀行存款的年利率保持不變，那么這種存款的年利率大約是多少？（精確到0.01%）若本金為a元，某種存款的年利率x，則一年到期應(yīng)得稅后利息為ax(120%)元,應(yīng)得稅后本息為a1x(120%)元；若沒取出再存一年（自動轉(zhuǎn)存），到期應(yīng)得稅后本息為a1x(120%)2元.解：解：設(shè)這種存款的年利率為x，根據(jù)題意，得 450500 x(120%)1 x (120%)461整理，得 320 x2760 x110解得,符合題意; 不合題意，舍去.答：這種存款的年利率大約是1.44%

10、.217607604 320 110.0144 1.44%2 320 x 227607604 320 1102 320 x 今天，我們主要通過分析幾個(gè)例題，看到列一元二次方程解應(yīng)用題的一般步驟及注意事項(xiàng)。首先，要適當(dāng)?shù)丶僭O(shè)未知數(shù)，這一步非常關(guān)鍵，往往影響后面解方程的計(jì)算量；再仔細(xì)分析題意，列出方程，解方程，得到方程的解；這時(shí)一定要注意檢驗(yàn)方程的解是否符合實(shí)際意義，不符合實(shí)際意義的解要舍去；最后答題。對于帶有單位的應(yīng)用題，如面積問題，在假設(shè)、答題中要帶著單位，中間過程不需要單位。)WMCrh7(WLBrg6(VLAqg5*VKAqf5&UKzpf4&UJzoe4%TJyod3%S

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27、Aqg5*UKApf5&UJzpe4&TJzoe3%TIyod3$SIxnd2$SHxmc2!RHwmc1!QGwlb1#QFvla0#PFvka+ZPEuk9+YOEtj9-YODti8-XNDsi8)XMCsh7)WMBrh6(WLBrg6*VLAqg5*VKApf5&UKzpe4&TJzoe4%TIyod3%SIxnd2$SHxnc2!RHwmc1!RGwlb1#QGvla0#PFvka0ZPEuk9+ZOEuj9-VKAqf5*UKApf4&UJzpe4%TJyoe3%TIynd3$SIxnd2$RHxmc2!RGwmb1!QGwlb0#QFvl

28、a0ZPFuka+ZPEuj9+YOEtj9-YNDti8-XNCsi7)XMCsh7(WMBrh6(VLBqg6*VLAqf5*UKApf5&UJzpe4&TJyoe3%TIyod3$SIxnd2$SHxmc2!RHwmb1!QGwlb1#QFvla0#P$2cnxHS$3dnyIS%3doyJT%4eozJU&4fpzKU&5fqAKV*5gqBLV(6grBLW(7hrCMW)7hsCNX)8isDNY-8jtDOY-9juEOZ+9kuFPZ0akvFP#0blvGQ#1blwGR!1cmwHR$2cnxHS$2dnyIS%3doyIT%4eozJT&a

29、mp;4fpzKU&5fpAKV*5gqALV(6grBLW(6hrCMW)7hsCMX)8isDNX-8jtDOY-9jtEOZ+9kuEPZ+akvFP#0alvGQ#1blwGQ!1cmwHR!2cnxHS$2dnxIS%3doyIT%3eozJT&4epzKU&5fpAKU*5gqALV*6grBLW(6hrBMW)7hsCMX)7isDNX-8itDOY-9jtEOY+9kuEPZ+akuFP#0alvFQ#1blwGQ!1bmwHR!2cmxHS$2dnxIS$3doyIT%3eoyJT&4epzJU&5fpAKU*5fqALV*6gqBLV

30、(6hrBMW(7hsCMX)7isCNX-8itDNY-9jtEOY+9juEPZ+akuFPZ0alvFQ#0blwGQ!1bmwGR!2cmxHR$2dnxIS$3dnyIT%3eoyJT%4epzJU&4fpAKU*5fqAKV*6gqBLV(6grBMW(7hrCMX)7isCNX)8itDNY-8jtEOY+9juEOZ+akuFPZ0akvFQ#0blvGQ!1bmwGR!1cmxHR$2cnxHS$3dnyIS%3eoyJT%4eozJU&4fpzKU*5fqAKV*5gqBLV(6grBLW(7hrCMW)7isCNX)8isDNY-8jtDOY+9juEOZ

31、+9kuFPZ0akvFP#0blvGQ#1bmwGR!1cmwHR$2cnxHS$2dnyIS%3doyJT%4eozJT&4fpzKU&5fqAKV*5gqALV(6grBLW(6hrCMW)7hsCNX)8isDNX-8jtDOY-9jtEOZ+9kuEPZ0akvFP#0alvGQ#1blwGR!1cmwHR!2cnxHS$2dnxIS%3doyIT%4eozJT&4epzKU&5fpAKU*5gqALV*6grBLW(6hrBMW)7hsCMX)8isDNX-8itDOY-9jtEOY+9kuEPZ+akvFP#0alvFQ#1blwGQ!1cmwHR

32、!2cmxHS$2dnxIS$3doyIT%3eozJT&4epzJU&5fpAKU*5fqALV*6gqBLW(6hrBMW(7hsCMX)7isDNX-8itDNY-9jtEOY+9juEPZ+akuFP#0alvFQ#0blwGQ!1bmwGR!2cmxHR$2dnxIS$3dnyIT%3eoyJT&4epzJU&4fpAKU*5fqAKV*6gqBLV(6hrBMW(7hrCMX)7isCNX-8itDNY-8jtEOY+9jxHS$2dnyIS%3doyIT%4eozJT&4fpzKU&5fpAKV*5gqALV*6grBLW(6hrC

33、MW)7hsCMX)8isDNX-8itDOY-9jtEOZ+9kuEPZ+akvFP#0alvGQ#1blwGQ!1cmwHR!2cmxHS$2dnxIS%3doyIT%3eozJT&4epzJU&5fpAKU*5gqALV*6gqBLW(6hrBMW)7hsCMX)7isDNX-8itDNY-9jtEOY+9kuEPZ+akuFP#0alvFQ#1blwGQ!1bmwHR!2cmxHR$2dnxIS$3doyIT%3eoyJT&4epzJU&4fpAKU*5fqALV*6gqBLV(6hrBMW(7hsCMX)7isCNX-8itDNY-8jtEOY+9ju

34、EPZ+akuFPZ0alvFQ#0blvGQ!1bmwGR!2cmxHR$2cnxIS$3dnyIT%3eoyJT%4epzJU&4fpzKU*5fqAKV*6gqBLV(6grBMW(7hrCMX)7isCNX)8itDNY-8jtDOY+9juEOZ+akuFPZ0akvFQ#0blvGQ#1bmwGR!1cmxHR$2cnxHS$3dnyIS%3eoyJT%4eozJU&4fpzKU&5fqAKV*5gqBLV(6grBLW(7hrCMW)7hsCNX)8isDNY-8jtDOY-9juEOZ+9kuFPZ0akvFP#0eoyJT&4epzJU&

35、;5fpAKU*5fqALV*6gqBLW(6hrBMW(7hsCMX)7isCNX-8itDNY-9jtEOY+9juEPZ+akuFP#0alvFQ#0blwGQ!1bmwGR!2cmxHR$2dnxIS$3dnyIT%3eoyJT%4epzJU&4fpAKU*5fqAKV*6gqBLV(6hrBMW(7hrCMX)7isCNX)8itDNY-8jtEOY+9juEOZ+akuFPZ0akvFQ#0blvGQ!1bmwGR!1cmxHR$2cnxIS$3dnyIS%3eoyJT%4eozJU&4fpzKU*5fqAKV*5gqBLV(6grBMW(7hrCMW)7isCNX

36、)8isDNY-8jtDOY+9juEOZ+9kuFPZ0akvFP#0blvGQ#1bmwGR!1cmwHR$2cnxHS$3dnyIS%3doyJT%4eozJT&4fpzKU&5fqAKV*5gqALV(6grBLW(6hrCMW)7hsCNX)8isDNX-8jtDOY-9juEOZ+9kuEPZ0akvFP#0alvGQ#1blwGR!1cmwHR!2cnxHS$2dnyIS%3doyIT%4eozJT&4epzKU&5fpAKV*5gqALV*6grBLW(6hrBMW)7hsCMX)8isDNX-8itDOY-9jtEOZ+9kuEPZ+akvFP

37、#0alvFQ#1blwGQ!1cmwHR!2cmxHS$2dnxIS$3doyIT%3eozJT&4epzJU&5fpAKU*5gqALV*6gqBLW(6hrBMW(7hsCMX)7isDNX-8itDNY-9jtEOY+9juEPZ+akuFP#0alvFQ#0blwGQ!1bmwHR!2cmxHR$2dnxIS$3dnyIT%3eoyJT&4epzJU&4fpAKU*5fqALV*6gqBLV(6hrBMW(7hrCMX)7isCNX-8itDNY-8jtEOY+9juES$2dnyIS%3doyIT%4eozJT&4fpzKU&5fpA

38、KV*5gqALV*6grBLW(6hrCMW)7hsCMX)8isDNX-8jtDOY-9jtEOZ+9kuEPZ+akvFP#0alvGQ#1blwGQ!1cmwHR!2cnxHS$2dnxIS%3doyIT%3eozJT&4epzKU&5fpAKU*5gqALV*6gqBLW(6hrBMW)7hsCMX)7isDNX-8itDOY-9jtEOY+9kuEPZ+akuFP#0alvFQ#1blwGQ!1bmwHR!2cmxHR$2dnxIS$3doyIT%3eoyJT&4epzJU&5fpAKU*5fqALV*6gqBLV(6hrBMW(7hsCMX)7is

39、CNX-8itDNY-8jtEOY+9juEPZ+anyIS%3doyJT%4eozJT&4fpzKU&5fqAKV*5gqALV(6grBLW(6hrCMW)7hsCNX)8isDNX-8jtDOY-9jtEOZ+9kuEPZ0akvFP#0alvGQ#1blwGR!1cmwHR!2cnxHS$2dnxIS%3doyIT%4eozJT&4epzKU&5fpAKU*5gqALV*6grBLW(6hrBMW)7hsCMX)8isDNX-8itDOY-9jtEOY+9kuEPZ+akvFP#0alvFQ#1blwGQ!1cmwHR!2cmxHS$2dnxIS$3do

40、yIT%3eozJT&4epzJU&5fpAKU*5fqALV*6gqBLW(6hrBMW(7hsCMX)7isDNX-8itDNY-9jtEOY+9juEPZ+akuFP#0alvFQ#0blwGQ!1bmwGR!2cmxHR$2dnxIS$3dnyIT%3eoyJT&4epzJU&4fpAKU*5fqAKV*6gqBLV(6hrBMW(7hrCMX)7isCNX)8itDNY-8jtEOY+9juEOZ+akuFPZ0alvFQ#0blvGQ!1bmwGR!1cmxHR$2cnxIS$3dnyIS%3eoyJT%4epzJU&4fpzKU*5fqAKV*5gqBLV(6grBMW(7hrCMW)7isCNX)8isDNY-8jtDOY+9juEOZ+9kxIS%3doyIT%3eozJT&4epzJU&5fpAKU*5gqALV*6gqBLW(6hrBMW)7hsCMX)7isDNX-8itDNY-9jtEOY+9kuEPZ+akuFP#0alvFQ#0blwGQ!1bmw