4y^2+24y-374=0 eching

$$4 y ^ { 2 } + 24 y - 374 = 0$$

y uchun yechish

$$y=\frac{\sqrt{410}}{2}-3\approx 7,124228366$$

$$y=-\frac{\sqrt{410}}{2}-3\approx -13,124228366$$

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.

$$4y^{2}+24y-374=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 24 ni b va -374 ni c bilan almashtiring.

$$y=\frac{-24±\sqrt{24^{2}-4\times 4\left(-374\right)}}{2\times 4}$$

24 kvadratini chiqarish.

$$y=\frac{-24±\sqrt{576-4\times 4\left(-374\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$y=\frac{-24±\sqrt{576-16\left(-374\right)}}{2\times 4}$$

-16 ni -374 marotabaga ko'paytirish.

$$y=\frac{-24±\sqrt{576+5984}}{2\times 4}$$

576 ni 5984 ga qo'shish.

$$y=\frac{-24±\sqrt{6560}}{2\times 4}$$

6560 ning kvadrat ildizini chiqarish.

$$y=\frac{-24±4\sqrt{410}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$y=\frac{-24±4\sqrt{410}}{8}$$

y=\frac{-24±4\sqrt{410}}{8} tenglamasini yeching, bunda ± musbat. -24 ni 4\sqrt{410} ga qo'shish.

$$y=\frac{4\sqrt{410}-24}{8}$$

-24+4\sqrt{410} ni 8 ga bo'lish.

$$y=\frac{\sqrt{410}}{2}-3$$

y=\frac{-24±4\sqrt{410}}{8} tenglamasini yeching, bunda ± manfiy. -24 dan 4\sqrt{410} ni ayirish.

$$y=\frac{-4\sqrt{410}-24}{8}$$

-24-4\sqrt{410} ni 8 ga bo'lish.

$$y=-\frac{\sqrt{410}}{2}-3$$

Tenglama yechildi.

$$y=\frac{\sqrt{410}}{2}-3 y=-\frac{\sqrt{410}}{2}-3$$

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.

$$4y^{2}+24y-374=0$$

374 ni tenglamaning ikkala tarafiga qo'shish.

$$4y^{2}+24y-374-\left(-374\right)=-\left(-374\right)$$

O‘zidan -374 ayirilsa 0 qoladi.

$$4y^{2}+24y=-\left(-374\right)$$

0 dan -374 ni ayirish.

$$4y^{2}+24y=374$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4y^{2}+24y}{4}=\frac{374}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$y^{2}+\frac{24}{4}y=\frac{374}{4}$$

24 ni 4 ga bo'lish.

$$y^{2}+6y=\frac{374}{4}$$

\frac{374}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y^{2}+6y=\frac{187}{2}$$

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

$$y^{2}+6y+3^{2}=\frac{187}{2}+3^{2}$$

3 kvadratini chiqarish.

$$y^{2}+6y+9=\frac{187}{2}+9$$

\frac{187}{2} ni 9 ga qo'shish.

$$y^{2}+6y+9=\frac{205}{2}$$

y^{2}+6y+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.

$$\left(y+3\right)^{2}=\frac{205}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{\frac{205}{2}}$$

Qisqartirish.

$$y+3=\frac{\sqrt{410}}{2} y+3=-\frac{\sqrt{410}}{2}$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$y=\frac{\sqrt{410}}{2}-3 y=-\frac{\sqrt{410}}{2}-3$$