(20-x)(100+20x)=2240 eching

$$(20-x)(100+20x)=2240$$

x uchun yechish

$$x = \frac{\sqrt{177} + 15}{2} \approx 14,152067348$$

$$x=\frac{15-\sqrt{177}}{2}\approx 0,847932652$$

Kvadrat tenglamalaridan foydalanish qoidalari

20-x ga 100+20x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+300x-20x^{2}=2240$$

Ikkala tarafdan 2240 ni ayirish.

$$2000+300x-20x^{2}-2240=0$$

-240 olish uchun 2000 dan 2240 ni ayirish.

$$-240+300x-20x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-20x^{2}+300x-240=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -20 ni a, 300 ni b va -240 ni c bilan almashtiring.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

300 kvadratini chiqarish.

$$x=\frac{-300±\sqrt{90000-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

-4 ni -20 marotabaga ko'paytirish.

$$x=\frac{-300±\sqrt{90000+80\left(-240\right)}}{2\left(-20\right)}$$

80 ni -240 marotabaga ko'paytirish.

$$x=\frac{-300±\sqrt{90000-19200}}{2\left(-20\right)}$$

90000 ni -19200 ga qo'shish.

$$x=\frac{-300±\sqrt{70800}}{2\left(-20\right)}$$

70800 ning kvadrat ildizini chiqarish.

$$x=\frac{-300±20\sqrt{177}}{2\left(-20\right)}$$

2 ni -20 marotabaga ko'paytirish.

$$x=\frac{-300±20\sqrt{177}}{-40}$$

x=\frac{-300±20\sqrt{177}}{-40} tenglamasini yeching, bunda ± musbat. -300 ni 20\sqrt{177} ga qo'shish.

$$x=\frac{20\sqrt{177}-300}{-40}$$

-300+20\sqrt{177} ni -40 ga bo'lish.

$$x=\frac{15-\sqrt{177}}{2}$$

x=\frac{-300±20\sqrt{177}}{-40} tenglamasini yeching, bunda ± manfiy. -300 dan 20\sqrt{177} ni ayirish.

$$x=\frac{-20\sqrt{177}-300}{-40}$$

-300-20\sqrt{177} ni -40 ga bo'lish.

$$x=\frac{\sqrt{177}+15}{2}$$

Tenglama yechildi.

$$x=\frac{15-\sqrt{177}}{2} x=\frac{\sqrt{177}+15}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

20-x ga 100+20x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+300x-20x^{2}=2240$$

Ikkala tarafdan 2000 ni ayirish.

$$300x-20x^{2}=2240-2000$$

240 olish uchun 2240 dan 2000 ni ayirish.

$$300x-20x^{2}=240$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-20x^{2}+300x=240$$

Ikki tarafini -20 ga bo‘ling.

$$\frac{-20x^{2}+300x}{-20}=\frac{240}{-20}$$

-20 ga bo'lish -20 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{300}{-20}x=\frac{240}{-20}$$

300 ni -20 ga bo'lish.

$$x^{2}-15x=\frac{240}{-20}$$

240 ni -20 ga bo'lish.

$$x^{2}-15x=-12$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-12+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-12+\frac{225}{4}$$

-12 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{177}{4}$$

x^{2}-15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{177}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{177}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{\sqrt{177}}{2} x-\frac{15}{2}=-\frac{\sqrt{177}}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{177}+15}{2} x=\frac{15-\sqrt{177}}{2}$$