-2x^2+2x+15=0 eching

$$- 2 x ^ { 2 } + 2 x + 15 = 0$$

x uchun yechish

$$x = \frac{\sqrt{31} + 1}{2} \approx 3,283882181$$

$$x=\frac{1-\sqrt{31}}{2}\approx -2,283882181$$

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.

$$-2x^{2}+2x+15=0$$

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

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

2 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+8\times 15}}{2\left(-2\right)}$$

8 ni 15 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+120}}{2\left(-2\right)}$$

4 ni 120 ga qo'shish.

$$x=\frac{-2±\sqrt{124}}{2\left(-2\right)}$$

124 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{31}}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{31}}{-4}$$

x=\frac{-2±2\sqrt{31}}{-4} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{31} ga qo'shish.

$$x=\frac{2\sqrt{31}-2}{-4}$$

-2+2\sqrt{31} ni -4 ga bo'lish.

$$x=\frac{1-\sqrt{31}}{2}$$

x=\frac{-2±2\sqrt{31}}{-4} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{31} ni ayirish.

$$x=\frac{-2\sqrt{31}-2}{-4}$$

-2-2\sqrt{31} ni -4 ga bo'lish.

$$x=\frac{\sqrt{31}+1}{2}$$

Tenglama yechildi.

$$x=\frac{1-\sqrt{31}}{2} x=\frac{\sqrt{31}+1}{2}$$

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.

$$-2x^{2}+2x+15=0$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$-2x^{2}+2x+15-15=-15$$

O‘zidan 15 ayirilsa 0 qoladi.

$$-2x^{2}+2x=-15$$

Ikki tarafini -2 ga bo‘ling.

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

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

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

2 ni -2 ga bo'lish.

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

-15 ni -2 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$x^{2}-x+\frac{1}{4}=\frac{15}{2}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{2} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-x+\frac{1}{4}=\frac{31}{4}$$

x^{2}-x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{31}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{31}{4}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{31}}{2} x-\frac{1}{2}=-\frac{\sqrt{31}}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{31}+1}{2} x=\frac{1-\sqrt{31}}{2}$$