69x^2+1157x-6760=0 eching

$$69 x ^ { 2 } + 1157 x - 6760 = 0$$

x uchun yechish

$$x = \frac{13 \sqrt{18961} - 1157}{138} \approx 4,587581962$$

$$x=\frac{-13\sqrt{18961}-1157}{138}\approx -21,355697904$$

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.

$$69x^{2}+1157x-6760=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 69 ni a, 1157 ni b va -6760 ni c bilan almashtiring.

$$x=\frac{-1157±\sqrt{1157^{2}-4\times 69\left(-6760\right)}}{2\times 69}$$

1157 kvadratini chiqarish.

$$x=\frac{-1157±\sqrt{1338649-4\times 69\left(-6760\right)}}{2\times 69}$$

-4 ni 69 marotabaga ko'paytirish.

$$x=\frac{-1157±\sqrt{1338649-276\left(-6760\right)}}{2\times 69}$$

-276 ni -6760 marotabaga ko'paytirish.

$$x=\frac{-1157±\sqrt{1338649+1865760}}{2\times 69}$$

1338649 ni 1865760 ga qo'shish.

$$x=\frac{-1157±\sqrt{3204409}}{2\times 69}$$

3204409 ning kvadrat ildizini chiqarish.

$$x=\frac{-1157±13\sqrt{18961}}{2\times 69}$$

2 ni 69 marotabaga ko'paytirish.

$$x=\frac{-1157±13\sqrt{18961}}{138}$$

x=\frac{-1157±13\sqrt{18961}}{138} tenglamasini yeching, bunda ± musbat. -1157 ni 13\sqrt{18961} ga qo'shish.

$$x=\frac{13\sqrt{18961}-1157}{138}$$

x=\frac{-1157±13\sqrt{18961}}{138} tenglamasini yeching, bunda ± manfiy. -1157 dan 13\sqrt{18961} ni ayirish.

$$x=\frac{-13\sqrt{18961}-1157}{138}$$

Tenglama yechildi.

$$x=\frac{13\sqrt{18961}-1157}{138} x=\frac{-13\sqrt{18961}-1157}{138}$$

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.

$$69x^{2}+1157x-6760=0$$

6760 ni tenglamaning ikkala tarafiga qo'shish.

$$69x^{2}+1157x-6760-\left(-6760\right)=-\left(-6760\right)$$

O‘zidan -6760 ayirilsa 0 qoladi.

$$69x^{2}+1157x=-\left(-6760\right)$$

0 dan -6760 ni ayirish.

$$69x^{2}+1157x=6760$$

Ikki tarafini 69 ga bo‘ling.

$$\frac{69x^{2}+1157x}{69}=\frac{6760}{69}$$

69 ga bo'lish 69 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1157}{69}x=\frac{6760}{69}$$

\frac{1157}{69} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1157}{138} olish uchun. Keyin, \frac{1157}{138} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1157}{69}x+\left(\frac{1157}{138}\right)^{2}=\frac{6760}{69}+\left(\frac{1157}{138}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1157}{138} kvadratini chiqarish.

$$x^{2}+\frac{1157}{69}x+\frac{1338649}{19044}=\frac{6760}{69}+\frac{1338649}{19044}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{6760}{69} ni \frac{1338649}{19044} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1157}{69}x+\frac{1338649}{19044}=\frac{3204409}{19044}$$

x^{2}+\frac{1157}{69}x+\frac{1338649}{19044} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1157}{138}\right)^{2}=\frac{3204409}{19044}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1157}{138}\right)^{2}}=\sqrt{\frac{3204409}{19044}}$$

Qisqartirish.

$$x+\frac{1157}{138}=\frac{13\sqrt{18961}}{138} x+\frac{1157}{138}=-\frac{13\sqrt{18961}}{138}$$

Tenglamaning ikkala tarafidan \frac{1157}{138} ni ayirish.

$$x=\frac{13\sqrt{18961}-1157}{138} x=\frac{-13\sqrt{18961}-1157}{138}$$