48x^2-52x-26=0 eching

$$48 x ^ { 2 } - 52 x - 26 = 0$$

x uchun yechish

$$x = \frac{\sqrt{481} + 13}{24} \approx 1,455488008$$

$$x=\frac{13-\sqrt{481}}{24}\approx -0,372154675$$

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.

$$48x^{2}-52x-26=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 48 ni a, -52 ni b va -26 ni c bilan almashtiring.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 48\left(-26\right)}}{2\times 48}$$

-52 kvadratini chiqarish.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 48\left(-26\right)}}{2\times 48}$$

-4 ni 48 marotabaga ko'paytirish.

$$x=\frac{-\left(-52\right)±\sqrt{2704-192\left(-26\right)}}{2\times 48}$$

-192 ni -26 marotabaga ko'paytirish.

$$x=\frac{-\left(-52\right)±\sqrt{2704+4992}}{2\times 48}$$

2704 ni 4992 ga qo'shish.

$$x=\frac{-\left(-52\right)±\sqrt{7696}}{2\times 48}$$

7696 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-52\right)±4\sqrt{481}}{2\times 48}$$

-52 ning teskarisi 52 ga teng.

$$x=\frac{52±4\sqrt{481}}{2\times 48}$$

2 ni 48 marotabaga ko'paytirish.

$$x=\frac{52±4\sqrt{481}}{96}$$

x=\frac{52±4\sqrt{481}}{96} tenglamasini yeching, bunda ± musbat. 52 ni 4\sqrt{481} ga qo'shish.

$$x=\frac{4\sqrt{481}+52}{96}$$

52+4\sqrt{481} ni 96 ga bo'lish.

$$x=\frac{\sqrt{481}+13}{24}$$

x=\frac{52±4\sqrt{481}}{96} tenglamasini yeching, bunda ± manfiy. 52 dan 4\sqrt{481} ni ayirish.

$$x=\frac{52-4\sqrt{481}}{96}$$

52-4\sqrt{481} ni 96 ga bo'lish.

$$x=\frac{13-\sqrt{481}}{24}$$

Tenglama yechildi.

$$x=\frac{\sqrt{481}+13}{24} x=\frac{13-\sqrt{481}}{24}$$

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.

$$48x^{2}-52x-26=0$$

26 ni tenglamaning ikkala tarafiga qo'shish.

$$48x^{2}-52x-26-\left(-26\right)=-\left(-26\right)$$

O‘zidan -26 ayirilsa 0 qoladi.

$$48x^{2}-52x=-\left(-26\right)$$

0 dan -26 ni ayirish.

$$48x^{2}-52x=26$$

Ikki tarafini 48 ga bo‘ling.

$$\frac{48x^{2}-52x}{48}=\frac{26}{48}$$

48 ga bo'lish 48 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{52}{48}\right)x=\frac{26}{48}$$

\frac{-52}{48} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{13}{12}x=\frac{26}{48}$$

\frac{26}{48} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{13}{12}x=\frac{13}{24}$$

-\frac{13}{12} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{24} olish uchun. Keyin, -\frac{13}{24} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{13}{12}x+\left(-\frac{13}{24}\right)^{2}=\frac{13}{24}+\left(-\frac{13}{24}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{24} kvadratini chiqarish.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{13}{24}+\frac{169}{576}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{13}{24} ni \frac{169}{576} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{481}{576}$$

x^{2}-\frac{13}{12}x+\frac{169}{576} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{13}{24}\right)^{2}=\frac{481}{576}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{24}\right)^{2}}=\sqrt{\frac{481}{576}}$$

Qisqartirish.

$$x-\frac{13}{24}=\frac{\sqrt{481}}{24} x-\frac{13}{24}=-\frac{\sqrt{481}}{24}$$

\frac{13}{24} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{481}+13}{24} x=\frac{13-\sqrt{481}}{24}$$