2y^2+10y=-9 eching

$$2 y ^ { 2 } + 10 y = - 9$$

y uchun yechish

$$y=\frac{\sqrt{7}-5}{2}\approx -1,177124344$$

$$y=\frac{-\sqrt{7}-5}{2}\approx -3,822875656$$

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.

$$2y^{2}+10y=-9$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$2y^{2}+10y-\left(-9\right)=-9-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$2y^{2}+10y-\left(-9\right)=0$$

0 dan -9 ni ayirish.

$$2y^{2}+10y+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 10 ni b va 9 ni c bilan almashtiring.

$$y=\frac{-10±\sqrt{10^{2}-4\times 2\times 9}}{2\times 2}$$

10 kvadratini chiqarish.

$$y=\frac{-10±\sqrt{100-4\times 2\times 9}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$y=\frac{-10±\sqrt{100-8\times 9}}{2\times 2}$$

-8 ni 9 marotabaga ko'paytirish.

$$y=\frac{-10±\sqrt{100-72}}{2\times 2}$$

100 ni -72 ga qo'shish.

$$y=\frac{-10±\sqrt{28}}{2\times 2}$$

28 ning kvadrat ildizini chiqarish.

$$y=\frac{-10±2\sqrt{7}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$y=\frac{-10±2\sqrt{7}}{4}$$

y=\frac{-10±2\sqrt{7}}{4} tenglamasini yeching, bunda ± musbat. -10 ni 2\sqrt{7} ga qo'shish.

$$y=\frac{2\sqrt{7}-10}{4}$$

-10+2\sqrt{7} ni 4 ga bo'lish.

$$y=\frac{\sqrt{7}-5}{2}$$

y=\frac{-10±2\sqrt{7}}{4} tenglamasini yeching, bunda ± manfiy. -10 dan 2\sqrt{7} ni ayirish.

$$y=\frac{-2\sqrt{7}-10}{4}$$

-10-2\sqrt{7} ni 4 ga bo'lish.

$$y=\frac{-\sqrt{7}-5}{2}$$

Tenglama yechildi.

$$y=\frac{\sqrt{7}-5}{2} y=\frac{-\sqrt{7}-5}{2}$$

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.

$$2y^{2}+10y=-9$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2y^{2}+10y}{2}=-\frac{9}{2}$$

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

$$y^{2}+\frac{10}{2}y=-\frac{9}{2}$$

10 ni 2 ga bo'lish.

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

5 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{2} olish uchun. Keyin, \frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}+5y+\left(\frac{5}{2}\right)^{2}=-\frac{9}{2}+\left(\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$y^{2}+5y+\frac{25}{4}=-\frac{9}{2}+\frac{25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{2} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}+5y+\frac{25}{4}=\frac{7}{4}$$

y^{2}+5y+\frac{25}{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.

$$\left(y+\frac{5}{2}\right)^{2}=\frac{7}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+\frac{5}{2}\right)^{2}}=\sqrt{\frac{7}{4}}$$

Qisqartirish.

$$y+\frac{5}{2}=\frac{\sqrt{7}}{2} y+\frac{5}{2}=-\frac{\sqrt{7}}{2}$$

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$y=\frac{\sqrt{7}-5}{2} y=\frac{-\sqrt{7}-5}{2}$$