4y^2-13y+36=0 eching

$$4 y ^ { 2 } - 13 y + 36 = 0$$

y uchun yechish

$$y=\frac{13+\sqrt{407}i}{8}\approx 1,625+2,521780125i$$

$$y=\frac{-\sqrt{407}i+13}{8}\approx 1,625-2,521780125i$$

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.

$$4y^{2}-13y+36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -13 ni b va 36 ni c bilan almashtiring.

$$y=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 4\times 36}}{2\times 4}$$

-13 kvadratini chiqarish.

$$y=\frac{-\left(-13\right)±\sqrt{169-4\times 4\times 36}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$y=\frac{-\left(-13\right)±\sqrt{169-16\times 36}}{2\times 4}$$

-16 ni 36 marotabaga ko'paytirish.

$$y=\frac{-\left(-13\right)±\sqrt{169-576}}{2\times 4}$$

169 ni -576 ga qo'shish.

$$y=\frac{-\left(-13\right)±\sqrt{-407}}{2\times 4}$$

-407 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-13\right)±\sqrt{407}i}{2\times 4}$$

-13 ning teskarisi 13 ga teng.

$$y=\frac{13±\sqrt{407}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$y=\frac{13±\sqrt{407}i}{8}$$

y=\frac{13±\sqrt{407}i}{8} tenglamasini yeching, bunda ± musbat. 13 ni i\sqrt{407} ga qo'shish.

$$y=\frac{13+\sqrt{407}i}{8}$$

y=\frac{13±\sqrt{407}i}{8} tenglamasini yeching, bunda ± manfiy. 13 dan i\sqrt{407} ni ayirish.

$$y=\frac{-\sqrt{407}i+13}{8}$$

Tenglama yechildi.

$$y=\frac{13+\sqrt{407}i}{8} y=\frac{-\sqrt{407}i+13}{8}$$

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.

$$4y^{2}-13y+36=0$$

Tenglamaning ikkala tarafidan 36 ni ayirish.

$$4y^{2}-13y+36-36=-36$$

O‘zidan 36 ayirilsa 0 qoladi.

$$4y^{2}-13y=-36$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4y^{2}-13y}{4}=-\frac{36}{4}$$

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

$$y^{2}-\frac{13}{4}y=-\frac{36}{4}$$

-36 ni 4 ga bo'lish.

$$y^{2}-\frac{13}{4}y=-9$$

-\frac{13}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{8} olish uchun. Keyin, -\frac{13}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-\frac{13}{4}y+\left(-\frac{13}{8}\right)^{2}=-9+\left(-\frac{13}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{8} kvadratini chiqarish.

$$y^{2}-\frac{13}{4}y+\frac{169}{64}=-9+\frac{169}{64}$$

-9 ni \frac{169}{64} ga qo'shish.

$$y^{2}-\frac{13}{4}y+\frac{169}{64}=-\frac{407}{64}$$

y^{2}-\frac{13}{4}y+\frac{169}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y-\frac{13}{8}\right)^{2}=-\frac{407}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{13}{8}\right)^{2}}=\sqrt{-\frac{407}{64}}$$

Qisqartirish.

$$y-\frac{13}{8}=\frac{\sqrt{407}i}{8} y-\frac{13}{8}=-\frac{\sqrt{407}i}{8}$$

\frac{13}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{13+\sqrt{407}i}{8} y=\frac{-\sqrt{407}i+13}{8}$$