2x^2+7x-6=0 eching

$$2 x ^ { 2 } + 7 x - 6 = 0$$

x uchun yechish

$$x=\frac{\sqrt{97}-7}{4}\approx 0,71221445$$

$$x=\frac{-\sqrt{97}-7}{4}\approx -4,21221445$$

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.

$$2x^{2}+7x-6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 7 ni b va -6 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-6\right)}}{2\times 2}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-6\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-8\left(-6\right)}}{2\times 2}$$

-8 ni -6 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+48}}{2\times 2}$$

49 ni 48 ga qo'shish.

$$x=\frac{-7±\sqrt{97}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{97}}{4}$$

x=\frac{-7±\sqrt{97}}{4} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{97} ga qo'shish.

$$x=\frac{\sqrt{97}-7}{4}$$

x=\frac{-7±\sqrt{97}}{4} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{97} ni ayirish.

$$x=\frac{-\sqrt{97}-7}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{97}-7}{4} x=\frac{-\sqrt{97}-7}{4}$$

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.

$$2x^{2}+7x-6=0$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}+7x-6-\left(-6\right)=-\left(-6\right)$$

O‘zidan -6 ayirilsa 0 qoladi.

$$2x^{2}+7x=-\left(-6\right)$$

0 dan -6 ni ayirish.

$$2x^{2}+7x=6$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+7x}{2}=\frac{6}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{2}x=\frac{6}{2}$$

6 ni 2 ga bo'lish.

$$x^{2}+\frac{7}{2}x=3$$

\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{4} olish uchun. Keyin, \frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=3+\left(\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{4} kvadratini chiqarish.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=3+\frac{49}{16}$$

3 ni \frac{49}{16} ga qo'shish.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{97}{16}$$

x^{2}+\frac{7}{2}x+\frac{49}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{97}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{97}{16}}$$

Qisqartirish.

$$x+\frac{7}{4}=\frac{\sqrt{97}}{4} x+\frac{7}{4}=-\frac{\sqrt{97}}{4}$$

Tenglamaning ikkala tarafidan \frac{7}{4} ni ayirish.

$$x=\frac{\sqrt{97}-7}{4} x=\frac{-\sqrt{97}-7}{4}$$