(1)x^2-3x-2=0 eching

$$( 1 ) x ^ { 2 } - 3 x - 2 = 0$$

x uchun yechish

$$x = \frac{\sqrt{17} + 3}{2} \approx 3,561552813$$

$$x=\frac{3-\sqrt{17}}{2}\approx -0,561552813$$

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -3 ni b va -2 ni c bilan almashtiring.

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

-3 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+8}}{2}$$

9 ni 8 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{17}}{2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{17}}{2}$$

x=\frac{3±\sqrt{17}}{2} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}+3}{2}$$

x=\frac{3±\sqrt{17}}{2} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{17} ni ayirish.

$$x=\frac{3-\sqrt{17}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{17}+3}{2} x=\frac{3-\sqrt{17}}{2}$$

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.

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

2 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -2 ayirilsa 0 qoladi.

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

0 dan -2 ni ayirish.

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

-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.

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

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

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

2 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{17}{4}$$

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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{17}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{17}}{2} x-\frac{3}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}+3}{2} x=\frac{3-\sqrt{17}}{2}$$