-2y^2-6y+5=0 eching

$$- 2 y ^ { 2 } - 6 y + 5 = 0$$

y uchun yechish

$$y=\frac{\sqrt{19}-3}{2}\approx 0,679449472$$

$$y=\frac{-\sqrt{19}-3}{2}\approx -3,679449472$$

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.

$$-2y^{2}-6y+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, -6 ni b va 5 ni c bilan almashtiring.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

-6 kvadratini chiqarish.

$$y=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$y=\frac{-\left(-6\right)±\sqrt{36+8\times 5}}{2\left(-2\right)}$$

8 ni 5 marotabaga ko'paytirish.

$$y=\frac{-\left(-6\right)±\sqrt{36+40}}{2\left(-2\right)}$$

36 ni 40 ga qo'shish.

$$y=\frac{-\left(-6\right)±\sqrt{76}}{2\left(-2\right)}$$

76 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-6\right)±2\sqrt{19}}{2\left(-2\right)}$$

-6 ning teskarisi 6 ga teng.

$$y=\frac{6±2\sqrt{19}}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$y=\frac{6±2\sqrt{19}}{-4}$$

y=\frac{6±2\sqrt{19}}{-4} tenglamasini yeching, bunda ± musbat. 6 ni 2\sqrt{19} ga qo'shish.

$$y=\frac{2\sqrt{19}+6}{-4}$$

6+2\sqrt{19} ni -4 ga bo'lish.

$$y=\frac{-\sqrt{19}-3}{2}$$

y=\frac{6±2\sqrt{19}}{-4} tenglamasini yeching, bunda ± manfiy. 6 dan 2\sqrt{19} ni ayirish.

$$y=\frac{6-2\sqrt{19}}{-4}$$

6-2\sqrt{19} ni -4 ga bo'lish.

$$y=\frac{\sqrt{19}-3}{2}$$

Tenglama yechildi.

$$y=\frac{-\sqrt{19}-3}{2} y=\frac{\sqrt{19}-3}{2}$$

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.

$$-2y^{2}-6y+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$-2y^{2}-6y+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$-2y^{2}-6y=-5$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2y^{2}-6y}{-2}=-\frac{5}{-2}$$

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

$$y^{2}+\left(-\frac{6}{-2}\right)y=-\frac{5}{-2}$$

-6 ni -2 ga bo'lish.

$$y^{2}+3y=-\frac{5}{-2}$$

-5 ni -2 ga bo'lish.

$$y^{2}+3y=\frac{5}{2}$$

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

$$y^{2}+3y+\left(\frac{3}{2}\right)^{2}=\frac{5}{2}+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$y^{2}+3y+\frac{9}{4}=\frac{5}{2}+\frac{9}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}+3y+\frac{9}{4}=\frac{19}{4}$$

y^{2}+3y+\frac{9}{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.

$$\left(y+\frac{3}{2}\right)^{2}=\frac{19}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Qisqartirish.

$$y+\frac{3}{2}=\frac{\sqrt{19}}{2} y+\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$y=\frac{\sqrt{19}-3}{2} y=\frac{-\sqrt{19}-3}{2}$$