60x^2+588x-169=0 eching

$$60 x ^ { 2 } + 588 x - 169 = 0$$

x uchun yechish

$$x=\frac{2\sqrt{1509}}{15}-\frac{49}{10}\approx 0,27944656$$

$$x=-\frac{2\sqrt{1509}}{15}-\frac{49}{10}\approx -10,07944656$$

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.

$$60x^{2}+588x-169=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 60 ni a, 588 ni b va -169 ni c bilan almashtiring.

$$x=\frac{-588±\sqrt{588^{2}-4\times 60\left(-169\right)}}{2\times 60}$$

588 kvadratini chiqarish.

$$x=\frac{-588±\sqrt{345744-4\times 60\left(-169\right)}}{2\times 60}$$

-4 ni 60 marotabaga ko'paytirish.

$$x=\frac{-588±\sqrt{345744-240\left(-169\right)}}{2\times 60}$$

-240 ni -169 marotabaga ko'paytirish.

$$x=\frac{-588±\sqrt{345744+40560}}{2\times 60}$$

345744 ni 40560 ga qo'shish.

$$x=\frac{-588±\sqrt{386304}}{2\times 60}$$

386304 ning kvadrat ildizini chiqarish.

$$x=\frac{-588±16\sqrt{1509}}{2\times 60}$$

2 ni 60 marotabaga ko'paytirish.

$$x=\frac{-588±16\sqrt{1509}}{120}$$

x=\frac{-588±16\sqrt{1509}}{120} tenglamasini yeching, bunda ± musbat. -588 ni 16\sqrt{1509} ga qo'shish.

$$x=\frac{16\sqrt{1509}-588}{120}$$

-588+16\sqrt{1509} ni 120 ga bo'lish.

$$x=\frac{2\sqrt{1509}}{15}-\frac{49}{10}$$

x=\frac{-588±16\sqrt{1509}}{120} tenglamasini yeching, bunda ± manfiy. -588 dan 16\sqrt{1509} ni ayirish.

$$x=\frac{-16\sqrt{1509}-588}{120}$$

-588-16\sqrt{1509} ni 120 ga bo'lish.

$$x=-\frac{2\sqrt{1509}}{15}-\frac{49}{10}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{1509}}{15}-\frac{49}{10} x=-\frac{2\sqrt{1509}}{15}-\frac{49}{10}$$

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.

$$60x^{2}+588x-169=0$$

169 ni tenglamaning ikkala tarafiga qo'shish.

$$60x^{2}+588x-169-\left(-169\right)=-\left(-169\right)$$

O‘zidan -169 ayirilsa 0 qoladi.

$$60x^{2}+588x=-\left(-169\right)$$

0 dan -169 ni ayirish.

$$60x^{2}+588x=169$$

Ikki tarafini 60 ga bo‘ling.

$$\frac{60x^{2}+588x}{60}=\frac{169}{60}$$

60 ga bo'lish 60 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{588}{60}x=\frac{169}{60}$$

\frac{588}{60} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{49}{5}x=\frac{169}{60}$$

\frac{49}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{49}{10} olish uchun. Keyin, \frac{49}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{49}{5}x+\left(\frac{49}{10}\right)^{2}=\frac{169}{60}+\left(\frac{49}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{49}{10} kvadratini chiqarish.

$$x^{2}+\frac{49}{5}x+\frac{2401}{100}=\frac{169}{60}+\frac{2401}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{169}{60} ni \frac{2401}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{49}{5}x+\frac{2401}{100}=\frac{2012}{75}$$

x^{2}+\frac{49}{5}x+\frac{2401}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{49}{10}\right)^{2}=\frac{2012}{75}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{49}{10}\right)^{2}}=\sqrt{\frac{2012}{75}}$$

Qisqartirish.

$$x+\frac{49}{10}=\frac{2\sqrt{1509}}{15} x+\frac{49}{10}=-\frac{2\sqrt{1509}}{15}$$

Tenglamaning ikkala tarafidan \frac{49}{10} ni ayirish.

$$x=\frac{2\sqrt{1509}}{15}-\frac{49}{10} x=-\frac{2\sqrt{1509}}{15}-\frac{49}{10}$$