8x^2-24x-24=0 eching

$$8 { x }^{ 2 } -24x-24=0$$

x uchun yechish

$$x = \frac{\sqrt{21} + 3}{2} \approx 3,791287847$$

$$x=\frac{3-\sqrt{21}}{2}\approx -0,791287847$$

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.

$$8x^{2}-24x-24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -24 ni b va -24 ni c bilan almashtiring.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 8\left(-24\right)}}{2\times 8}$$

-24 kvadratini chiqarish.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 8\left(-24\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576-32\left(-24\right)}}{2\times 8}$$

-32 ni -24 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576+768}}{2\times 8}$$

576 ni 768 ga qo'shish.

$$x=\frac{-\left(-24\right)±\sqrt{1344}}{2\times 8}$$

1344 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-24\right)±8\sqrt{21}}{2\times 8}$$

-24 ning teskarisi 24 ga teng.

$$x=\frac{24±8\sqrt{21}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{24±8\sqrt{21}}{16}$$

x=\frac{24±8\sqrt{21}}{16} tenglamasini yeching, bunda ± musbat. 24 ni 8\sqrt{21} ga qo'shish.

$$x=\frac{8\sqrt{21}+24}{16}$$

24+8\sqrt{21} ni 16 ga bo'lish.

$$x=\frac{\sqrt{21}+3}{2}$$

x=\frac{24±8\sqrt{21}}{16} tenglamasini yeching, bunda ± manfiy. 24 dan 8\sqrt{21} ni ayirish.

$$x=\frac{24-8\sqrt{21}}{16}$$

24-8\sqrt{21} ni 16 ga bo'lish.

$$x=\frac{3-\sqrt{21}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{21}+3}{2} x=\frac{3-\sqrt{21}}{2}$$

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.

$$8x^{2}-24x-24=0$$

24 ni tenglamaning ikkala tarafiga qo'shish.

$$8x^{2}-24x-24-\left(-24\right)=-\left(-24\right)$$

O‘zidan -24 ayirilsa 0 qoladi.

$$8x^{2}-24x=-\left(-24\right)$$

0 dan -24 ni ayirish.

$$8x^{2}-24x=24$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}-24x}{8}=\frac{24}{8}$$

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

$$x^{2}+\left(-\frac{24}{8}\right)x=\frac{24}{8}$$

-24 ni 8 ga bo'lish.

$$x^{2}-3x=\frac{24}{8}$$

24 ni 8 ga bo'lish.

$$x^{2}-3x=3$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=3+\frac{9}{4}$$

3 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{21}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{21}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{21}}{2} x-\frac{3}{2}=-\frac{\sqrt{21}}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{21}+3}{2} x=\frac{3-\sqrt{21}}{2}$$