2x^2-8x-223=0 eching

$$2 x ^ { 2 } - 8 x - 223 = 0$$

x uchun yechish

$$x=\frac{\sqrt{462}}{2}+2\approx 12,74709263$$

$$x=-\frac{\sqrt{462}}{2}+2\approx -8,74709263$$

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.

$$2x^{2}-8x-223=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -8 ni b va -223 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-223\right)}}{2\times 2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-223\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-223\right)}}{2\times 2}$$

-8 ni -223 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+1784}}{2\times 2}$$

64 ni 1784 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{1848}}{2\times 2}$$

1848 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±2\sqrt{462}}{2\times 2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±2\sqrt{462}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{8±2\sqrt{462}}{4}$$

x=\frac{8±2\sqrt{462}}{4} tenglamasini yeching, bunda ± musbat. 8 ni 2\sqrt{462} ga qo'shish.

$$x=\frac{2\sqrt{462}+8}{4}$$

8+2\sqrt{462} ni 4 ga bo'lish.

$$x=\frac{\sqrt{462}}{2}+2$$

x=\frac{8±2\sqrt{462}}{4} tenglamasini yeching, bunda ± manfiy. 8 dan 2\sqrt{462} ni ayirish.

$$x=\frac{8-2\sqrt{462}}{4}$$

8-2\sqrt{462} ni 4 ga bo'lish.

$$x=-\frac{\sqrt{462}}{2}+2$$

Tenglama yechildi.

$$x=\frac{\sqrt{462}}{2}+2 x=-\frac{\sqrt{462}}{2}+2$$

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.

$$2x^{2}-8x-223=0$$

223 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-8x-223-\left(-223\right)=-\left(-223\right)$$

O‘zidan -223 ayirilsa 0 qoladi.

$$2x^{2}-8x=-\left(-223\right)$$

0 dan -223 ni ayirish.

$$2x^{2}-8x=223$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-8x}{2}=\frac{223}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{223}{2}$$

-8 ni 2 ga bo'lish.

$$x^{2}-4x=\frac{223}{2}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=\frac{223}{2}+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=\frac{223}{2}+4$$

\frac{223}{2} ni 4 ga qo'shish.

$$x^{2}-4x+4=\frac{231}{2}$$

x^{2}-4x+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.

$$\left(x-2\right)^{2}=\frac{231}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{231}{2}}$$

Qisqartirish.

$$x-2=\frac{\sqrt{462}}{2} x-2=-\frac{\sqrt{462}}{2}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{462}}{2}+2 x=-\frac{\sqrt{462}}{2}+2$$