(-y^2+3y+5=0) eching

$$(- { y }^{ 2 } +3y+5=0)$$

y uchun yechish

$$y = \frac{\sqrt{29} + 3}{2} \approx 4,192582404$$

$$y=\frac{3-\sqrt{29}}{2}\approx -1,192582404$$

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.

$$-y^{2}+3y+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 3 ni b va 5 ni c bilan almashtiring.

$$y=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

3 kvadratini chiqarish.

$$y=\frac{-3±\sqrt{9-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$y=\frac{-3±\sqrt{9+4\times 5}}{2\left(-1\right)}$$

4 ni 5 marotabaga ko'paytirish.

$$y=\frac{-3±\sqrt{9+20}}{2\left(-1\right)}$$

9 ni 20 ga qo'shish.

$$y=\frac{-3±\sqrt{29}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$y=\frac{-3±\sqrt{29}}{-2}$$

y=\frac{-3±\sqrt{29}}{-2} tenglamasini yeching, bunda ± musbat. -3 ni \sqrt{29} ga qo'shish.

$$y=\frac{\sqrt{29}-3}{-2}$$

-3+\sqrt{29} ni -2 ga bo'lish.

$$y=\frac{3-\sqrt{29}}{2}$$

y=\frac{-3±\sqrt{29}}{-2} tenglamasini yeching, bunda ± manfiy. -3 dan \sqrt{29} ni ayirish.

$$y=\frac{-\sqrt{29}-3}{-2}$$

-3-\sqrt{29} ni -2 ga bo'lish.

$$y=\frac{\sqrt{29}+3}{2}$$

Tenglama yechildi.

$$y=\frac{3-\sqrt{29}}{2} y=\frac{\sqrt{29}+3}{2}$$

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.

$$-y^{2}+3y+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

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

O‘zidan 5 ayirilsa 0 qoladi.

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

Ikki tarafini -1 ga bo‘ling.

$$\frac{-y^{2}+3y}{-1}=-\frac{5}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$y^{2}+\frac{3}{-1}y=-\frac{5}{-1}$$

3 ni -1 ga bo'lish.

$$y^{2}-3y=-\frac{5}{-1}$$

-5 ni -1 ga bo'lish.

$$y^{2}-3y=5$$

-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.

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

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

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

5 ni \frac{9}{4} ga qo'shish.

$$y^{2}-3y+\frac{9}{4}=\frac{29}{4}$$

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.

$$\left(y-\frac{3}{2}\right)^{2}=\frac{29}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{3}{2}\right)^{2}}=\sqrt{\frac{29}{4}}$$

Qisqartirish.

$$y-\frac{3}{2}=\frac{\sqrt{29}}{2} y-\frac{3}{2}=-\frac{\sqrt{29}}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{\sqrt{29}+3}{2} y=\frac{3-\sqrt{29}}{2}$$