2x^2+16x-1=0 eching

$$2 { x }^{ 2 } +16x-1=0$$

x uchun yechish

$$x=\frac{\sqrt{66}}{2}-4\approx 0,062019202$$

$$x=-\frac{\sqrt{66}}{2}-4\approx -8,062019202$$

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.

$$2x^{2}+16x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 16 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\times 2\left(-1\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-8\left(-1\right)}}{2\times 2}$$

-8 ni -1 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+8}}{2\times 2}$$

256 ni 8 ga qo'shish.

$$x=\frac{-16±\sqrt{264}}{2\times 2}$$

264 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±2\sqrt{66}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-16±2\sqrt{66}}{4}$$

x=\frac{-16±2\sqrt{66}}{4} tenglamasini yeching, bunda ± musbat. -16 ni 2\sqrt{66} ga qo'shish.

$$x=\frac{2\sqrt{66}-16}{4}$$

-16+2\sqrt{66} ni 4 ga bo'lish.

$$x=\frac{\sqrt{66}}{2}-4$$

x=\frac{-16±2\sqrt{66}}{4} tenglamasini yeching, bunda ± manfiy. -16 dan 2\sqrt{66} ni ayirish.

$$x=\frac{-2\sqrt{66}-16}{4}$$

-16-2\sqrt{66} ni 4 ga bo'lish.

$$x=-\frac{\sqrt{66}}{2}-4$$

Tenglama yechildi.

$$x=\frac{\sqrt{66}}{2}-4 x=-\frac{\sqrt{66}}{2}-4$$

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.

$$2x^{2}+16x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}+16x-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$2x^{2}+16x=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$2x^{2}+16x=1$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+16x}{2}=\frac{1}{2}$$

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

$$x^{2}+\frac{16}{2}x=\frac{1}{2}$$

16 ni 2 ga bo'lish.

$$x^{2}+8x=\frac{1}{2}$$

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

$$x^{2}+8x+4^{2}=\frac{1}{2}+4^{2}$$

4 kvadratini chiqarish.

$$x^{2}+8x+16=\frac{1}{2}+16$$

\frac{1}{2} ni 16 ga qo'shish.

$$x^{2}+8x+16=\frac{33}{2}$$

x^{2}+8x+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.

$$\left(x+4\right)^{2}=\frac{33}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{\frac{33}{2}}$$

Qisqartirish.

$$x+4=\frac{\sqrt{66}}{2} x+4=-\frac{\sqrt{66}}{2}$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$x=\frac{\sqrt{66}}{2}-4 x=-\frac{\sqrt{66}}{2}-4$$