9:6x^2-x=15 eching

$$9 : 6 x ^ { 2 } - x = 15$$

x uchun yechish

$$x = \frac{\sqrt{91} + 1}{3} \approx 3,513130671$$

$$x=\frac{1-\sqrt{91}}{3}\approx -2,846464005$$

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.

$$\frac{3}{2}x^{2}-x=15$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$\frac{3}{2}x^{2}-x-15=15-15$$

O‘zidan 15 ayirilsa 0 qoladi.

$$\frac{3}{2}x^{2}-x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{3}{2} ni a, -1 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{3}{2}\left(-15\right)}}{2\times \frac{3}{2}}$$

-4 ni \frac{3}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-6\left(-15\right)}}{2\times \frac{3}{2}}$$

-6 ni -15 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+90}}{2\times \frac{3}{2}}$$

1 ni 90 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{91}}{2\times \frac{3}{2}}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{91}}{2\times \frac{3}{2}}$$

2 ni \frac{3}{2} marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{91}}{3}$$

x=\frac{1±\sqrt{91}}{3} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{91} ga qo'shish.

$$x=\frac{\sqrt{91}+1}{3}$$

x=\frac{1±\sqrt{91}}{3} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{91} ni ayirish.

$$x=\frac{1-\sqrt{91}}{3}$$

Tenglama yechildi.

$$x=\frac{\sqrt{91}+1}{3} x=\frac{1-\sqrt{91}}{3}$$

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.

$$\frac{3}{2}x^{2}-x=15$$

Tenglamaning ikki tarafini \frac{3}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{3}{2}x^{2}-x}{\frac{3}{2}}=\frac{15}{\frac{3}{2}}$$

\frac{3}{2} ga bo'lish \frac{3}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{1}{\frac{3}{2}}\right)x=\frac{15}{\frac{3}{2}}$$

-1 ni \frac{3}{2} ga bo'lish -1 ga k'paytirish \frac{3}{2} ga qaytarish.

$$x^{2}-\frac{2}{3}x=\frac{15}{\frac{3}{2}}$$

15 ni \frac{3}{2} ga bo'lish 15 ga k'paytirish \frac{3}{2} ga qaytarish.

$$x^{2}-\frac{2}{3}x=10$$

-\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{3} olish uchun. Keyin, -\frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=10+\left(-\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=10+\frac{1}{9}$$

10 ni \frac{1}{9} ga qo'shish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{91}{9}$$

x^{2}-\frac{2}{3}x+\frac{1}{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.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{91}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{91}{9}}$$

Qisqartirish.

$$x-\frac{1}{3}=\frac{\sqrt{91}}{3} x-\frac{1}{3}=-\frac{\sqrt{91}}{3}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{91}+1}{3} x=\frac{1-\sqrt{91}}{3}$$