5x^2-48x-48=0 eching

$$5 { x }^{ 2 } -48x-48=0$$

x uchun yechish

$$x = \frac{4 \sqrt{51} + 24}{5} \approx 10,513142743$$

$$x=\frac{24-4\sqrt{51}}{5}\approx -0,913142743$$

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.

$$5x^{2}-48x-48=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -48 ni b va -48 ni c bilan almashtiring.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 5\left(-48\right)}}{2\times 5}$$

-48 kvadratini chiqarish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 5\left(-48\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-20\left(-48\right)}}{2\times 5}$$

-20 ni -48 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304+960}}{2\times 5}$$

2304 ni 960 ga qo'shish.

$$x=\frac{-\left(-48\right)±\sqrt{3264}}{2\times 5}$$

3264 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-48\right)±8\sqrt{51}}{2\times 5}$$

-48 ning teskarisi 48 ga teng.

$$x=\frac{48±8\sqrt{51}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{48±8\sqrt{51}}{10}$$

x=\frac{48±8\sqrt{51}}{10} tenglamasini yeching, bunda ± musbat. 48 ni 8\sqrt{51} ga qo'shish.

$$x=\frac{8\sqrt{51}+48}{10}$$

48+8\sqrt{51} ni 10 ga bo'lish.

$$x=\frac{4\sqrt{51}+24}{5}$$

x=\frac{48±8\sqrt{51}}{10} tenglamasini yeching, bunda ± manfiy. 48 dan 8\sqrt{51} ni ayirish.

$$x=\frac{48-8\sqrt{51}}{10}$$

48-8\sqrt{51} ni 10 ga bo'lish.

$$x=\frac{24-4\sqrt{51}}{5}$$

Tenglama yechildi.

$$x=\frac{4\sqrt{51}+24}{5} x=\frac{24-4\sqrt{51}}{5}$$

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.

$$5x^{2}-48x-48=0$$

48 ni tenglamaning ikkala tarafiga qo'shish.

$$5x^{2}-48x-48-\left(-48\right)=-\left(-48\right)$$

O‘zidan -48 ayirilsa 0 qoladi.

$$5x^{2}-48x=-\left(-48\right)$$

0 dan -48 ni ayirish.

$$5x^{2}-48x=48$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-48x}{5}=\frac{48}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{48}{5}x=\frac{48}{5}$$

-\frac{48}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{24}{5} olish uchun. Keyin, -\frac{24}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{48}{5}x+\left(-\frac{24}{5}\right)^{2}=\frac{48}{5}+\left(-\frac{24}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{24}{5} kvadratini chiqarish.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{48}{5}+\frac{576}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{48}{5} ni \frac{576}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{816}{25}$$

x^{2}-\frac{48}{5}x+\frac{576}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{24}{5}\right)^{2}=\frac{816}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{24}{5}\right)^{2}}=\sqrt{\frac{816}{25}}$$

Qisqartirish.

$$x-\frac{24}{5}=\frac{4\sqrt{51}}{5} x-\frac{24}{5}=-\frac{4\sqrt{51}}{5}$$

\frac{24}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4\sqrt{51}+24}{5} x=\frac{24-4\sqrt{51}}{5}$$