x^2-4x^2-8x+7=0 eching

$$x ^ { 2 } - 4 x ^ { 2 } - 8 x + 7 = 0$$

x uchun yechish

$$x=\frac{\sqrt{37}-4}{3}\approx 0,694254177$$

$$x=\frac{-\sqrt{37}-4}{3}\approx -3,360920843$$

Kvadrat tenglamalaridan foydalanish qoidalari

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$-3x^{2}-8x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -8 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-3\right)\times 7}}{2\left(-3\right)}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-3\right)\times 7}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+12\times 7}}{2\left(-3\right)}$$

12 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+84}}{2\left(-3\right)}$$

64 ni 84 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{148}}{2\left(-3\right)}$$

148 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±2\sqrt{37}}{2\left(-3\right)}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±2\sqrt{37}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{8±2\sqrt{37}}{-6}$$

x=\frac{8±2\sqrt{37}}{-6} tenglamasini yeching, bunda ± musbat. 8 ni 2\sqrt{37} ga qo'shish.

$$x=\frac{2\sqrt{37}+8}{-6}$$

8+2\sqrt{37} ni -6 ga bo'lish.

$$x=\frac{-\sqrt{37}-4}{3}$$

x=\frac{8±2\sqrt{37}}{-6} tenglamasini yeching, bunda ± manfiy. 8 dan 2\sqrt{37} ni ayirish.

$$x=\frac{8-2\sqrt{37}}{-6}$$

8-2\sqrt{37} ni -6 ga bo'lish.

$$x=\frac{\sqrt{37}-4}{3}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{37}-4}{3} x=\frac{\sqrt{37}-4}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$-3x^{2}-8x+7=0$$

Ikkala tarafdan 7 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-3x^{2}-8x=-7$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-8x}{-3}=-\frac{7}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{8}{-3}\right)x=-\frac{7}{-3}$$

-8 ni -3 ga bo'lish.

$$x^{2}+\frac{8}{3}x=-\frac{7}{-3}$$

-7 ni -3 ga bo'lish.

$$x^{2}+\frac{8}{3}x=\frac{7}{3}$$

\frac{8}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{3} olish uchun. Keyin, \frac{4}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{7}{3}+\left(\frac{4}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{3} kvadratini chiqarish.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{7}{3}+\frac{16}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{3} ni \frac{16}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{37}{9}$$

x^{2}+\frac{8}{3}x+\frac{16}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{37}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{37}{9}}$$

Qisqartirish.

$$x+\frac{4}{3}=\frac{\sqrt{37}}{3} x+\frac{4}{3}=-\frac{\sqrt{37}}{3}$$

Tenglamaning ikkala tarafidan \frac{4}{3} ni ayirish.

$$x=\frac{\sqrt{37}-4}{3} x=\frac{-\sqrt{37}-4}{3}$$