8x^2+x-3=0 eching

$$8 x ^ { 2 } + x - 3 = 0$$

x uchun yechish

$$x=\frac{\sqrt{97}-1}{16}\approx 0,553053613$$

$$x=\frac{-\sqrt{97}-1}{16}\approx -0,678053613$$

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.

$$8x^{2}+x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 1 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\times 8\left(-3\right)}}{2\times 8}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\times 8\left(-3\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-32\left(-3\right)}}{2\times 8}$$

-32 ni -3 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+96}}{2\times 8}$$

1 ni 96 ga qo'shish.

$$x=\frac{-1±\sqrt{97}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{97}}{16}$$

x=\frac{-1±\sqrt{97}}{16} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{97} ga qo'shish.

$$x=\frac{\sqrt{97}-1}{16}$$

x=\frac{-1±\sqrt{97}}{16} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{97} ni ayirish.

$$x=\frac{-\sqrt{97}-1}{16}$$

Tenglama yechildi.

$$x=\frac{\sqrt{97}-1}{16} x=\frac{-\sqrt{97}-1}{16}$$

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.

$$8x^{2}+x-3=0$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$8x^{2}+x-3-\left(-3\right)=-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$8x^{2}+x=-\left(-3\right)$$

0 dan -3 ni ayirish.

$$8x^{2}+x=3$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}+x}{8}=\frac{3}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1}{8}x=\frac{3}{8}$$

\frac{1}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{16} olish uchun. Keyin, \frac{1}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{8}x+\left(\frac{1}{16}\right)^{2}=\frac{3}{8}+\left(\frac{1}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{16} kvadratini chiqarish.

$$x^{2}+\frac{1}{8}x+\frac{1}{256}=\frac{3}{8}+\frac{1}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{8} ni \frac{1}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{8}x+\frac{1}{256}=\frac{97}{256}$$

x^{2}+\frac{1}{8}x+\frac{1}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{16}\right)^{2}=\frac{97}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{16}\right)^{2}}=\sqrt{\frac{97}{256}}$$

Qisqartirish.

$$x+\frac{1}{16}=\frac{\sqrt{97}}{16} x+\frac{1}{16}=-\frac{\sqrt{97}}{16}$$

Tenglamaning ikkala tarafidan \frac{1}{16} ni ayirish.

$$x=\frac{\sqrt{97}-1}{16} x=\frac{-\sqrt{97}-1}{16}$$