(5+a)^2+a=8+a eching

$$( 5 + a ) ^ { 2 } + a = 8 + a$$

a uchun yechish

$$a=2\sqrt{2}-5\approx -2,171572875$$

$$a=-2\sqrt{2}-5\approx -7,828427125$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(5+a\right)^{2} kengaytirilishi uchun ishlating.

$$25+10a+a^{2}+a=8+a$$

11a ni olish uchun 10a va a ni birlashtirish.

$$25+11a+a^{2}=8+a$$

Ikkala tarafdan 8 ni ayirish.

$$25+11a+a^{2}-8=a$$

17 olish uchun 25 dan 8 ni ayirish.

$$17+11a+a^{2}=a$$

Ikkala tarafdan a ni ayirish.

$$17+11a+a^{2}-a=0$$

10a ni olish uchun 11a va -a ni birlashtirish.

$$17+10a+a^{2}=0$$

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.

$$a^{2}+10a+17=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 10 ni b va 17 ni c bilan almashtiring.

$$a=\frac{-10±\sqrt{10^{2}-4\times 17}}{2}$$

10 kvadratini chiqarish.

$$a=\frac{-10±\sqrt{100-4\times 17}}{2}$$

-4 ni 17 marotabaga ko'paytirish.

$$a=\frac{-10±\sqrt{100-68}}{2}$$

100 ni -68 ga qo'shish.

$$a=\frac{-10±\sqrt{32}}{2}$$

32 ning kvadrat ildizini chiqarish.

$$a=\frac{-10±4\sqrt{2}}{2}$$

a=\frac{-10±4\sqrt{2}}{2} tenglamasini yeching, bunda ± musbat. -10 ni 4\sqrt{2} ga qo'shish.

$$a=\frac{4\sqrt{2}-10}{2}$$

-10+4\sqrt{2} ni 2 ga bo'lish.

$$a=2\sqrt{2}-5$$

a=\frac{-10±4\sqrt{2}}{2} tenglamasini yeching, bunda ± manfiy. -10 dan 4\sqrt{2} ni ayirish.

$$a=\frac{-4\sqrt{2}-10}{2}$$

-10-4\sqrt{2} ni 2 ga bo'lish.

$$a=-2\sqrt{2}-5$$

Tenglama yechildi.

$$a=2\sqrt{2}-5 a=-2\sqrt{2}-5$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(5+a\right)^{2} kengaytirilishi uchun ishlating.

$$25+10a+a^{2}+a=8+a$$

11a ni olish uchun 10a va a ni birlashtirish.

$$25+11a+a^{2}=8+a$$

Ikkala tarafdan a ni ayirish.

$$25+11a+a^{2}-a=8$$

10a ni olish uchun 11a va -a ni birlashtirish.

$$25+10a+a^{2}=8$$

Ikkala tarafdan 25 ni ayirish.

$$10a+a^{2}=8-25$$

-17 olish uchun 8 dan 25 ni ayirish.

$$10a+a^{2}=-17$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$a^{2}+10a=-17$$

10 ni bo‘lish, x shartining koeffitsienti, 2 ga 5 olish uchun. Keyin, 5 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}+10a+5^{2}=-17+5^{2}$$

5 kvadratini chiqarish.

$$a^{2}+10a+25=-17+25$$

-17 ni 25 ga qo'shish.

$$a^{2}+10a+25=8$$

a^{2}+10a+25 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(a+5\right)^{2}=8$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a+5\right)^{2}}=\sqrt{8}$$

Qisqartirish.

$$a+5=2\sqrt{2} a+5=-2\sqrt{2}$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$a=2\sqrt{2}-5 a=-2\sqrt{2}-5$$