k^2-k=8 eching

$$k ^ { 2 } - k = 8$$

k uchun yechish

$$k = \frac{\sqrt{33} + 1}{2} \approx 3,372281323$$

$$k=\frac{1-\sqrt{33}}{2}\approx -2,372281323$$

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.

$$k^{2}-k=8$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$k^{2}-k-8=8-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$k^{2}-k-8=0$$

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

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

-4 ni -8 marotabaga ko'paytirish.

$$k=\frac{-\left(-1\right)±\sqrt{1+32}}{2}$$

1 ni 32 ga qo'shish.

$$k=\frac{-\left(-1\right)±\sqrt{33}}{2}$$

-1 ning teskarisi 1 ga teng.

$$k=\frac{1±\sqrt{33}}{2}$$

k=\frac{1±\sqrt{33}}{2} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{33} ga qo'shish.

$$k=\frac{\sqrt{33}+1}{2}$$

k=\frac{1±\sqrt{33}}{2} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{33} ni ayirish.

$$k=\frac{1-\sqrt{33}}{2}$$

Tenglama yechildi.

$$k=\frac{\sqrt{33}+1}{2} k=\frac{1-\sqrt{33}}{2}$$

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.

$$k^{2}-k=8$$

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

$$k^{2}-k+\left(-\frac{1}{2}\right)^{2}=8+\left(-\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

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

8 ni \frac{1}{4} ga qo'shish.

$$k^{2}-k+\frac{1}{4}=\frac{33}{4}$$

k^{2}-k+\frac{1}{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.

$$\left(k-\frac{1}{2}\right)^{2}=\frac{33}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k-\frac{1}{2}\right)^{2}}=\sqrt{\frac{33}{4}}$$

Qisqartirish.

$$k-\frac{1}{2}=\frac{\sqrt{33}}{2} k-\frac{1}{2}=-\frac{\sqrt{33}}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$k=\frac{\sqrt{33}+1}{2} k=\frac{1-\sqrt{33}}{2}$$