2*(a^2-9)+a=15 eching

$$2 \cdot ( a ^ { 2 } - 9 ) + a = 15$$

a uchun yechish

$$a = \frac{\sqrt{265} - 1}{4} \approx 3,819705149$$

$$a=\frac{-\sqrt{265}-1}{4}\approx -4,319705149$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 ga a^{2}-9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2a^{2}-18+a=15$$

Ikkala tarafdan 15 ni ayirish.

$$2a^{2}-18+a-15=0$$

-33 olish uchun -18 dan 15 ni ayirish.

$$2a^{2}-33+a=0$$

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.

$$2a^{2}+a-33=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 1 ni b va -33 ni c bilan almashtiring.

$$a=\frac{-1±\sqrt{1^{2}-4\times 2\left(-33\right)}}{2\times 2}$$

1 kvadratini chiqarish.

$$a=\frac{-1±\sqrt{1-4\times 2\left(-33\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$a=\frac{-1±\sqrt{1-8\left(-33\right)}}{2\times 2}$$

-8 ni -33 marotabaga ko'paytirish.

$$a=\frac{-1±\sqrt{1+264}}{2\times 2}$$

1 ni 264 ga qo'shish.

$$a=\frac{-1±\sqrt{265}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$a=\frac{-1±\sqrt{265}}{4}$$

a=\frac{-1±\sqrt{265}}{4} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{265} ga qo'shish.

$$a=\frac{\sqrt{265}-1}{4}$$

a=\frac{-1±\sqrt{265}}{4} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{265} ni ayirish.

$$a=\frac{-\sqrt{265}-1}{4}$$

Tenglama yechildi.

$$a=\frac{\sqrt{265}-1}{4} a=\frac{-\sqrt{265}-1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

2 ga a^{2}-9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2a^{2}-18+a=15$$

18 ni ikki tarafga qo’shing.

$$2a^{2}+a=15+18$$

33 olish uchun 15 va 18'ni qo'shing.

$$2a^{2}+a=33$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2a^{2}+a}{2}=\frac{33}{2}$$

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

$$a^{2}+\frac{1}{2}a=\frac{33}{2}$$

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}+\frac{1}{2}a+\left(\frac{1}{4}\right)^{2}=\frac{33}{2}+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$a^{2}+\frac{1}{2}a+\frac{1}{16}=\frac{33}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{33}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$a^{2}+\frac{1}{2}a+\frac{1}{16}=\frac{265}{16}$$

a^{2}+\frac{1}{2}a+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(a+\frac{1}{4}\right)^{2}=\frac{265}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a+\frac{1}{4}\right)^{2}}=\sqrt{\frac{265}{16}}$$

Qisqartirish.

$$a+\frac{1}{4}=\frac{\sqrt{265}}{4} a+\frac{1}{4}=-\frac{\sqrt{265}}{4}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$a=\frac{\sqrt{265}-1}{4} a=\frac{-\sqrt{265}-1}{4}$$