2x^2+9x-1=0 eching

$$2 x ^ { 2 } + 9 x - 1 = 0$$

x uchun yechish

$$x=\frac{\sqrt{89}-9}{4}\approx 0,108495283$$

$$x=\frac{-\sqrt{89}-9}{4}\approx -4,608495283$$

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.

$$2x^{2}+9x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 9 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\times 2\left(-1\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-8\left(-1\right)}}{2\times 2}$$

-8 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+8}}{2\times 2}$$

81 ni 8 ga qo'shish.

$$x=\frac{-9±\sqrt{89}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{89}}{4}$$

x=\frac{-9±\sqrt{89}}{4} tenglamasini yeching, bunda ± musbat. -9 ni \sqrt{89} ga qo'shish.

$$x=\frac{\sqrt{89}-9}{4}$$

x=\frac{-9±\sqrt{89}}{4} tenglamasini yeching, bunda ± manfiy. -9 dan \sqrt{89} ni ayirish.

$$x=\frac{-\sqrt{89}-9}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{89}-9}{4} x=\frac{-\sqrt{89}-9}{4}$$

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.

$$2x^{2}+9x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}+9x-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$2x^{2}+9x=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$2x^{2}+9x=1$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+9x}{2}=\frac{1}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{9}{2}x=\frac{1}{2}$$

\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{4} olish uchun. Keyin, \frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=\frac{1}{2}+\left(\frac{9}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{4} kvadratini chiqarish.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{1}{2}+\frac{81}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{81}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{89}{16}$$

x^{2}+\frac{9}{2}x+\frac{81}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{89}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{89}{16}}$$

Qisqartirish.

$$x+\frac{9}{4}=\frac{\sqrt{89}}{4} x+\frac{9}{4}=-\frac{\sqrt{89}}{4}$$

Tenglamaning ikkala tarafidan \frac{9}{4} ni ayirish.

$$x=\frac{\sqrt{89}-9}{4} x=\frac{-\sqrt{89}-9}{4}$$