-9x^2+18x+68=0 eching

$$- 9 x ^ { 2 } + 18 x + 68 = 0$$

x uchun yechish

$$x=\frac{\sqrt{77}}{3}+1\approx 3,924988129$$

$$x=-\frac{\sqrt{77}}{3}+1\approx -1,924988129$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$-9x^{2}+18x+68=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, 18 ni b va 68 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324+36\times 68}}{2\left(-9\right)}$$

36 ni 68 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324+2448}}{2\left(-9\right)}$$

324 ni 2448 ga qo'shish.

$$x=\frac{-18±\sqrt{2772}}{2\left(-9\right)}$$

2772 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±6\sqrt{77}}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{-18±6\sqrt{77}}{-18}$$

x=\frac{-18±6\sqrt{77}}{-18} tenglamasini yeching, bunda ± musbat. -18 ni 6\sqrt{77} ga qo'shish.

$$x=\frac{6\sqrt{77}-18}{-18}$$

-18+6\sqrt{77} ni -18 ga bo'lish.

$$x=-\frac{\sqrt{77}}{3}+1$$

x=\frac{-18±6\sqrt{77}}{-18} tenglamasini yeching, bunda ± manfiy. -18 dan 6\sqrt{77} ni ayirish.

$$x=\frac{-6\sqrt{77}-18}{-18}$$

-18-6\sqrt{77} ni -18 ga bo'lish.

$$x=\frac{\sqrt{77}}{3}+1$$

Tenglama yechildi.

$$x=-\frac{\sqrt{77}}{3}+1 x=\frac{\sqrt{77}}{3}+1$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$-9x^{2}+18x+68=0$$

Tenglamaning ikkala tarafidan 68 ni ayirish.

$$-9x^{2}+18x+68-68=-68$$

O‘zidan 68 ayirilsa 0 qoladi.

$$-9x^{2}+18x=-68$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}+18x}{-9}=-\frac{68}{-9}$$

-9 ga bo'lish -9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{18}{-9}x=-\frac{68}{-9}$$

18 ni -9 ga bo'lish.

$$x^{2}-2x=-\frac{68}{-9}$$

-68 ni -9 ga bo'lish.

$$x^{2}-2x=\frac{68}{9}$$

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=\frac{68}{9}+1$$

\frac{68}{9} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{77}{9}$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=\frac{77}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{77}{9}}$$

Qisqartirish.

$$x-1=\frac{\sqrt{77}}{3} x-1=-\frac{\sqrt{77}}{3}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{77}}{3}+1 x=-\frac{\sqrt{77}}{3}+1$$