x^2-2x-9=0 eching

$$x ^ { 2 } - 2 x - 9 = 0$$

x uchun yechish

$$x=\sqrt{10}+1\approx 4,16227766$$

$$x=1-\sqrt{10}\approx -2,16227766$$

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.

$$x^{2}-2x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -2 ni b va -9 ni c bilan almashtiring.

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

-2 kvadratini chiqarish.

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

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+36}}{2}$$

4 ni 36 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{40}}{2}$$

40 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{10}}{2}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{10}}{2}$$

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

$$x=\frac{2\sqrt{10}+2}{2}$$

2+2\sqrt{10} ni 2 ga bo'lish.

$$x=\sqrt{10}+1$$

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

$$x=\frac{2-2\sqrt{10}}{2}$$

2-2\sqrt{10} ni 2 ga bo'lish.

$$x=1-\sqrt{10}$$

Tenglama yechildi.

$$x=\sqrt{10}+1 x=1-\sqrt{10}$$

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.

$$x^{2}-2x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -9 ayirilsa 0 qoladi.

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

0 dan -9 ni ayirish.

$$x^{2}-2x=9$$

-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.

$$x^{2}-2x+1=9+1$$

9 ni 1 ga qo'shish.

$$x^{2}-2x+1=10$$

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.

$$\left(x-1\right)^{2}=10$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{10}$$

Qisqartirish.

$$x-1=\sqrt{10} x-1=-\sqrt{10}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{10}+1 x=1-\sqrt{10}$$