9x^2+150x-119=0 eching

$$9 { x }^{ 2 } +150x-119=0$$

x uchun yechish

$$x=\frac{2\sqrt{186}-25}{3}\approx 0,758787798$$

$$x=\frac{-2\sqrt{186}-25}{3}\approx -17,425454465$$

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.

$$9x^{2}+150x-119=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, 150 ni b va -119 ni c bilan almashtiring.

$$x=\frac{-150±\sqrt{150^{2}-4\times 9\left(-119\right)}}{2\times 9}$$

150 kvadratini chiqarish.

$$x=\frac{-150±\sqrt{22500-4\times 9\left(-119\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-150±\sqrt{22500-36\left(-119\right)}}{2\times 9}$$

-36 ni -119 marotabaga ko'paytirish.

$$x=\frac{-150±\sqrt{22500+4284}}{2\times 9}$$

22500 ni 4284 ga qo'shish.

$$x=\frac{-150±\sqrt{26784}}{2\times 9}$$

26784 ning kvadrat ildizini chiqarish.

$$x=\frac{-150±12\sqrt{186}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{-150±12\sqrt{186}}{18}$$

x=\frac{-150±12\sqrt{186}}{18} tenglamasini yeching, bunda ± musbat. -150 ni 12\sqrt{186} ga qo'shish.

$$x=\frac{12\sqrt{186}-150}{18}$$

-150+12\sqrt{186} ni 18 ga bo'lish.

$$x=\frac{2\sqrt{186}-25}{3}$$

x=\frac{-150±12\sqrt{186}}{18} tenglamasini yeching, bunda ± manfiy. -150 dan 12\sqrt{186} ni ayirish.

$$x=\frac{-12\sqrt{186}-150}{18}$$

-150-12\sqrt{186} ni 18 ga bo'lish.

$$x=\frac{-2\sqrt{186}-25}{3}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{186}-25}{3} x=\frac{-2\sqrt{186}-25}{3}$$

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.

$$9x^{2}+150x-119=0$$

119 ni tenglamaning ikkala tarafiga qo'shish.

$$9x^{2}+150x-119-\left(-119\right)=-\left(-119\right)$$

O‘zidan -119 ayirilsa 0 qoladi.

$$9x^{2}+150x=-\left(-119\right)$$

0 dan -119 ni ayirish.

$$9x^{2}+150x=119$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}+150x}{9}=\frac{119}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{150}{9}x=\frac{119}{9}$$

\frac{150}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{50}{3}x=\frac{119}{9}$$

\frac{50}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{25}{3} olish uchun. Keyin, \frac{25}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{50}{3}x+\left(\frac{25}{3}\right)^{2}=\frac{119}{9}+\left(\frac{25}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{25}{3} kvadratini chiqarish.

$$x^{2}+\frac{50}{3}x+\frac{625}{9}=\frac{119+625}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{119}{9} ni \frac{625}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{50}{3}x+\frac{625}{9}=\frac{248}{3}$$

x^{2}+\frac{50}{3}x+\frac{625}{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.

$$\left(x+\frac{25}{3}\right)^{2}=\frac{248}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{25}{3}\right)^{2}}=\sqrt{\frac{248}{3}}$$

Qisqartirish.

$$x+\frac{25}{3}=\frac{2\sqrt{186}}{3} x+\frac{25}{3}=-\frac{2\sqrt{186}}{3}$$

Tenglamaning ikkala tarafidan \frac{25}{3} ni ayirish.

$$x=\frac{2\sqrt{186}-25}{3} x=\frac{-2\sqrt{186}-25}{3}$$