(-50+6x)(-50+16x)=left(-50+14xright)^2 eching

$$(-50+6x)(-50+16x)= { \left(-50+14x \right) }^{ 2 }$$

x uchun yechish

$$x=3$$

$$x=0$$

Faktorlash qadamlari

-50+6x ga -50+16x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2500-1100x+96x^{2}=\left(-50+14x\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-50+14x\right)^{2} kengaytirilishi uchun ishlating.

$$2500-1100x+96x^{2}=2500-1400x+196x^{2}$$

Ikkala tarafdan 2500 ni ayirish.

$$2500-1100x+96x^{2}-2500=-1400x+196x^{2}$$

0 olish uchun 2500 dan 2500 ni ayirish.

$$-1100x+96x^{2}=-1400x+196x^{2}$$

1400x ni ikki tarafga qo’shing.

$$-1100x+96x^{2}+1400x=196x^{2}$$

300x ni olish uchun -1100x va 1400x ni birlashtirish.

$$300x+96x^{2}=196x^{2}$$

Ikkala tarafdan 196x^{2} ni ayirish.

$$300x+96x^{2}-196x^{2}=0$$

-100x^{2} ni olish uchun 96x^{2} va -196x^{2} ni birlashtirish.

$$300x-100x^{2}=0$$

x omili.

$$x\left(300-100x\right)=0$$

Tenglamani yechish uchun x=0 va 300-100x=0 ni yeching.

$$x=0 x=3$$

Kvadrat tenglamalaridan foydalanish qoidalari

-50+6x ga -50+16x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2500-1100x+96x^{2}=\left(-50+14x\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-50+14x\right)^{2} kengaytirilishi uchun ishlating.

$$2500-1100x+96x^{2}=2500-1400x+196x^{2}$$

Ikkala tarafdan 2500 ni ayirish.

$$2500-1100x+96x^{2}-2500=-1400x+196x^{2}$$

0 olish uchun 2500 dan 2500 ni ayirish.

$$-1100x+96x^{2}=-1400x+196x^{2}$$

1400x ni ikki tarafga qo’shing.

$$-1100x+96x^{2}+1400x=196x^{2}$$

300x ni olish uchun -1100x va 1400x ni birlashtirish.

$$300x+96x^{2}=196x^{2}$$

Ikkala tarafdan 196x^{2} ni ayirish.

$$300x+96x^{2}-196x^{2}=0$$

-100x^{2} ni olish uchun 96x^{2} va -196x^{2} ni birlashtirish.

$$300x-100x^{2}=0$$

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.

$$-100x^{2}+300x=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -100 ni a, 300 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-300±\sqrt{300^{2}}}{2\left(-100\right)}$$

300^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-300±300}{2\left(-100\right)}$$

2 ni -100 marotabaga ko'paytirish.

$$x=\frac{-300±300}{-200}$$

x=\frac{-300±300}{-200} tenglamasini yeching, bunda ± musbat. -300 ni 300 ga qo'shish.

$$x=\frac{0}{-200}$$

0 ni -200 ga bo'lish.

$$x=0$$

x=\frac{-300±300}{-200} tenglamasini yeching, bunda ± manfiy. -300 dan 300 ni ayirish.

$$x=-\frac{600}{-200}$$

-600 ni -200 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=0 x=3$$

Kvadratni yakunlash bo'yicha qadamlar

-50+6x ga -50+16x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2500-1100x+96x^{2}=\left(-50+14x\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-50+14x\right)^{2} kengaytirilishi uchun ishlating.

$$2500-1100x+96x^{2}=2500-1400x+196x^{2}$$

1400x ni ikki tarafga qo’shing.

$$2500-1100x+96x^{2}+1400x=2500+196x^{2}$$

300x ni olish uchun -1100x va 1400x ni birlashtirish.

$$2500+300x+96x^{2}=2500+196x^{2}$$

Ikkala tarafdan 196x^{2} ni ayirish.

$$2500+300x+96x^{2}-196x^{2}=2500$$

-100x^{2} ni olish uchun 96x^{2} va -196x^{2} ni birlashtirish.

$$2500+300x-100x^{2}=2500$$

Ikkala tarafdan 2500 ni ayirish.

$$300x-100x^{2}=2500-2500$$

0 olish uchun 2500 dan 2500 ni ayirish.

$$300x-100x^{2}=0$$

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

$$-100x^{2}+300x=0$$

Ikki tarafini -100 ga bo‘ling.

$$\frac{-100x^{2}+300x}{-100}=\frac{0}{-100}$$

-100 ga bo'lish -100 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{300}{-100}x=\frac{0}{-100}$$

300 ni -100 ga bo'lish.

$$x^{2}-3x=\frac{0}{-100}$$

0 ni -100 ga bo'lish.

$$x^{2}-3x=0$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=\left(-\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{3}{2} x-\frac{3}{2}=-\frac{3}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=0$$