(x-4)(28-x)-16=103 eching

$$( x - 4 ) ( 28 - x ) - 16 = 103$$

x uchun yechish

$$x=11$$

$$x=21$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-4 ga 28-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$32x-x^{2}-112-16=103$$

-128 olish uchun -112 dan 16 ni ayirish.

$$32x-x^{2}-128=103$$

Ikkala tarafdan 103 ni ayirish.

$$32x-x^{2}-128-103=0$$

-231 olish uchun -128 dan 103 ni ayirish.

$$32x-x^{2}-231=0$$

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.

$$-x^{2}+32x-231=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 32 ni b va -231 ni c bilan almashtiring.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-1\right)\left(-231\right)}}{2\left(-1\right)}$$

32 kvadratini chiqarish.

$$x=\frac{-32±\sqrt{1024-4\left(-1\right)\left(-231\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-32±\sqrt{1024+4\left(-231\right)}}{2\left(-1\right)}$$

4 ni -231 marotabaga ko'paytirish.

$$x=\frac{-32±\sqrt{1024-924}}{2\left(-1\right)}$$

1024 ni -924 ga qo'shish.

$$x=\frac{-32±\sqrt{100}}{2\left(-1\right)}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-32±10}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-32±10}{-2}$$

x=\frac{-32±10}{-2} tenglamasini yeching, bunda ± musbat. -32 ni 10 ga qo'shish.

$$x=-\frac{22}{-2}$$

-22 ni -2 ga bo'lish.

$$x=11$$

x=\frac{-32±10}{-2} tenglamasini yeching, bunda ± manfiy. -32 dan 10 ni ayirish.

$$x=-\frac{42}{-2}$$

-42 ni -2 ga bo'lish.

$$x=21$$

Tenglama yechildi.

$$x=11 x=21$$

Kvadratni yakunlash bo'yicha qadamlar

x-4 ga 28-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$32x-x^{2}-112-16=103$$

-128 olish uchun -112 dan 16 ni ayirish.

$$32x-x^{2}-128=103$$

128 ni ikki tarafga qo’shing.

$$32x-x^{2}=103+128$$

231 olish uchun 103 va 128'ni qo'shing.

$$32x-x^{2}=231$$

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

$$-x^{2}+32x=231$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+32x}{-1}=\frac{231}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{32}{-1}x=\frac{231}{-1}$$

32 ni -1 ga bo'lish.

$$x^{2}-32x=\frac{231}{-1}$$

231 ni -1 ga bo'lish.

$$x^{2}-32x=-231$$

-32 ni bo‘lish, x shartining koeffitsienti, 2 ga -16 olish uchun. Keyin, -16 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-32x+\left(-16\right)^{2}=-231+\left(-16\right)^{2}$$

-16 kvadratini chiqarish.

$$x^{2}-32x+256=-231+256$$

-231 ni 256 ga qo'shish.

$$x^{2}-32x+256=25$$

x^{2}-32x+256 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-16\right)^{2}=25$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{25}$$

Qisqartirish.

$$x-16=5 x-16=-5$$

16 ni tenglamaning ikkala tarafiga qo'shish.

$$x=21 x=11$$