576x^2+286x-135=0 eching

$$576 { x }^{ 2 } +286x-135=0$$

x uchun yechish

$$x=\frac{\sqrt{98209}-143}{576}\approx 0,295804083$$

$$x=\frac{-\sqrt{98209}-143}{576}\approx -0,792331861$$

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.

$$576x^{2}+286x-135=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 576 ni a, 286 ni b va -135 ni c bilan almashtiring.

$$x=\frac{-286±\sqrt{286^{2}-4\times 576\left(-135\right)}}{2\times 576}$$

286 kvadratini chiqarish.

$$x=\frac{-286±\sqrt{81796-4\times 576\left(-135\right)}}{2\times 576}$$

-4 ni 576 marotabaga ko'paytirish.

$$x=\frac{-286±\sqrt{81796-2304\left(-135\right)}}{2\times 576}$$

-2304 ni -135 marotabaga ko'paytirish.

$$x=\frac{-286±\sqrt{81796+311040}}{2\times 576}$$

81796 ni 311040 ga qo'shish.

$$x=\frac{-286±\sqrt{392836}}{2\times 576}$$

392836 ning kvadrat ildizini chiqarish.

$$x=\frac{-286±2\sqrt{98209}}{2\times 576}$$

2 ni 576 marotabaga ko'paytirish.

$$x=\frac{-286±2\sqrt{98209}}{1152}$$

x=\frac{-286±2\sqrt{98209}}{1152} tenglamasini yeching, bunda ± musbat. -286 ni 2\sqrt{98209} ga qo'shish.

$$x=\frac{2\sqrt{98209}-286}{1152}$$

-286+2\sqrt{98209} ni 1152 ga bo'lish.

$$x=\frac{\sqrt{98209}-143}{576}$$

x=\frac{-286±2\sqrt{98209}}{1152} tenglamasini yeching, bunda ± manfiy. -286 dan 2\sqrt{98209} ni ayirish.

$$x=\frac{-2\sqrt{98209}-286}{1152}$$

-286-2\sqrt{98209} ni 1152 ga bo'lish.

$$x=\frac{-\sqrt{98209}-143}{576}$$

Tenglama yechildi.

$$x=\frac{\sqrt{98209}-143}{576} x=\frac{-\sqrt{98209}-143}{576}$$

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.

$$576x^{2}+286x-135=0$$

135 ni tenglamaning ikkala tarafiga qo'shish.

$$576x^{2}+286x-135-\left(-135\right)=-\left(-135\right)$$

O‘zidan -135 ayirilsa 0 qoladi.

$$576x^{2}+286x=-\left(-135\right)$$

0 dan -135 ni ayirish.

$$576x^{2}+286x=135$$

Ikki tarafini 576 ga bo‘ling.

$$\frac{576x^{2}+286x}{576}=\frac{135}{576}$$

576 ga bo'lish 576 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{286}{576}x=\frac{135}{576}$$

\frac{286}{576} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{143}{288}x=\frac{135}{576}$$

\frac{135}{576} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{143}{288}x=\frac{15}{64}$$

\frac{143}{288} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{143}{576} olish uchun. Keyin, \frac{143}{576} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{143}{288}x+\left(\frac{143}{576}\right)^{2}=\frac{15}{64}+\left(\frac{143}{576}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{143}{576} kvadratini chiqarish.

$$x^{2}+\frac{143}{288}x+\frac{20449}{331776}=\frac{15}{64}+\frac{20449}{331776}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{64} ni \frac{20449}{331776} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{143}{288}x+\frac{20449}{331776}=\frac{98209}{331776}$$

x^{2}+\frac{143}{288}x+\frac{20449}{331776} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{143}{576}\right)^{2}=\frac{98209}{331776}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{143}{576}\right)^{2}}=\sqrt{\frac{98209}{331776}}$$

Qisqartirish.

$$x+\frac{143}{576}=\frac{\sqrt{98209}}{576} x+\frac{143}{576}=-\frac{\sqrt{98209}}{576}$$

Tenglamaning ikkala tarafidan \frac{143}{576} ni ayirish.

$$x=\frac{\sqrt{98209}-143}{576} x=\frac{-\sqrt{98209}-143}{576}$$