5y^2-90y+54=0 eching

$$5 { y }^{ 2 } -90y+54 = 0$$

y uchun yechish

$$y=\frac{3\sqrt{195}}{5}+9\approx 17,378544026$$

$$y=-\frac{3\sqrt{195}}{5}+9\approx 0,621455974$$

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.

$$5y^{2}-90y+54=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -90 ni b va 54 ni c bilan almashtiring.

$$y=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 5\times 54}}{2\times 5}$$

-90 kvadratini chiqarish.

$$y=\frac{-\left(-90\right)±\sqrt{8100-4\times 5\times 54}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$y=\frac{-\left(-90\right)±\sqrt{8100-20\times 54}}{2\times 5}$$

-20 ni 54 marotabaga ko'paytirish.

$$y=\frac{-\left(-90\right)±\sqrt{8100-1080}}{2\times 5}$$

8100 ni -1080 ga qo'shish.

$$y=\frac{-\left(-90\right)±\sqrt{7020}}{2\times 5}$$

7020 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-90\right)±6\sqrt{195}}{2\times 5}$$

-90 ning teskarisi 90 ga teng.

$$y=\frac{90±6\sqrt{195}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$y=\frac{90±6\sqrt{195}}{10}$$

y=\frac{90±6\sqrt{195}}{10} tenglamasini yeching, bunda ± musbat. 90 ni 6\sqrt{195} ga qo'shish.

$$y=\frac{6\sqrt{195}+90}{10}$$

90+6\sqrt{195} ni 10 ga bo'lish.

$$y=\frac{3\sqrt{195}}{5}+9$$

y=\frac{90±6\sqrt{195}}{10} tenglamasini yeching, bunda ± manfiy. 90 dan 6\sqrt{195} ni ayirish.

$$y=\frac{90-6\sqrt{195}}{10}$$

90-6\sqrt{195} ni 10 ga bo'lish.

$$y=-\frac{3\sqrt{195}}{5}+9$$

Tenglama yechildi.

$$y=\frac{3\sqrt{195}}{5}+9 y=-\frac{3\sqrt{195}}{5}+9$$

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.

$$5y^{2}-90y+54=0$$

Tenglamaning ikkala tarafidan 54 ni ayirish.

$$5y^{2}-90y+54-54=-54$$

O‘zidan 54 ayirilsa 0 qoladi.

$$5y^{2}-90y=-54$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5y^{2}-90y}{5}=-\frac{54}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$y^{2}+\left(-\frac{90}{5}\right)y=-\frac{54}{5}$$

-90 ni 5 ga bo'lish.

$$y^{2}-18y=-\frac{54}{5}$$

-18 ni bo‘lish, x shartining koeffitsienti, 2 ga -9 olish uchun. Keyin, -9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-18y+\left(-9\right)^{2}=-\frac{54}{5}+\left(-9\right)^{2}$$

-9 kvadratini chiqarish.

$$y^{2}-18y+81=-\frac{54}{5}+81$$

-\frac{54}{5} ni 81 ga qo'shish.

$$y^{2}-18y+81=\frac{351}{5}$$

y^{2}-18y+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y-9\right)^{2}=\frac{351}{5}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-9\right)^{2}}=\sqrt{\frac{351}{5}}$$

Qisqartirish.

$$y-9=\frac{3\sqrt{195}}{5} y-9=-\frac{3\sqrt{195}}{5}$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{3\sqrt{195}}{5}+9 y=-\frac{3\sqrt{195}}{5}+9$$