28x^2-8x-48=0 eching

$$28 x ^ { 2 } - 8 x - 48 = 0$$

x uchun yechish

$$x = \frac{\sqrt{85} + 1}{7} \approx 1,459934922$$

$$x=\frac{1-\sqrt{85}}{7}\approx -1,174220637$$

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.

$$28x^{2}-8x-48=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 28 ni a, -8 ni b va -48 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 28\left(-48\right)}}{2\times 28}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 28\left(-48\right)}}{2\times 28}$$

-4 ni 28 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-112\left(-48\right)}}{2\times 28}$$

-112 ni -48 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+5376}}{2\times 28}$$

64 ni 5376 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{5440}}{2\times 28}$$

5440 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±8\sqrt{85}}{2\times 28}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±8\sqrt{85}}{2\times 28}$$

2 ni 28 marotabaga ko'paytirish.

$$x=\frac{8±8\sqrt{85}}{56}$$

x=\frac{8±8\sqrt{85}}{56} tenglamasini yeching, bunda ± musbat. 8 ni 8\sqrt{85} ga qo'shish.

$$x=\frac{8\sqrt{85}+8}{56}$$

8+8\sqrt{85} ni 56 ga bo'lish.

$$x=\frac{\sqrt{85}+1}{7}$$

x=\frac{8±8\sqrt{85}}{56} tenglamasini yeching, bunda ± manfiy. 8 dan 8\sqrt{85} ni ayirish.

$$x=\frac{8-8\sqrt{85}}{56}$$

8-8\sqrt{85} ni 56 ga bo'lish.

$$x=\frac{1-\sqrt{85}}{7}$$

Tenglama yechildi.

$$x=\frac{\sqrt{85}+1}{7} x=\frac{1-\sqrt{85}}{7}$$

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.

$$28x^{2}-8x-48=0$$

48 ni tenglamaning ikkala tarafiga qo'shish.

$$28x^{2}-8x-48-\left(-48\right)=-\left(-48\right)$$

O‘zidan -48 ayirilsa 0 qoladi.

$$28x^{2}-8x=-\left(-48\right)$$

0 dan -48 ni ayirish.

$$28x^{2}-8x=48$$

Ikki tarafini 28 ga bo‘ling.

$$\frac{28x^{2}-8x}{28}=\frac{48}{28}$$

28 ga bo'lish 28 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{8}{28}\right)x=\frac{48}{28}$$

\frac{-8}{28} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{7}x=\frac{48}{28}$$

\frac{48}{28} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{7}x=\frac{12}{7}$$

-\frac{2}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{7} olish uchun. Keyin, -\frac{1}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{12}{7}+\left(-\frac{1}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{7} kvadratini chiqarish.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{12}{7}+\frac{1}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{12}{7} ni \frac{1}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{85}{49}$$

x^{2}-\frac{2}{7}x+\frac{1}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{7}\right)^{2}=\frac{85}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{\frac{85}{49}}$$

Qisqartirish.

$$x-\frac{1}{7}=\frac{\sqrt{85}}{7} x-\frac{1}{7}=-\frac{\sqrt{85}}{7}$$

\frac{1}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{85}+1}{7} x=\frac{1-\sqrt{85}}{7}$$