4a^2+102a-224=0 eching

$$4 a ^ { 2 } + 102 a - 224 = 0$$

a uchun yechish

$$a = \frac{\sqrt{3497} - 51}{4} \approx 2,033859442$$

$$a=\frac{-\sqrt{3497}-51}{4}\approx -27,533859442$$

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.

$$4a^{2}+102a-224=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 102 ni b va -224 ni c bilan almashtiring.

$$a=\frac{-102±\sqrt{102^{2}-4\times 4\left(-224\right)}}{2\times 4}$$

102 kvadratini chiqarish.

$$a=\frac{-102±\sqrt{10404-4\times 4\left(-224\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$a=\frac{-102±\sqrt{10404-16\left(-224\right)}}{2\times 4}$$

-16 ni -224 marotabaga ko'paytirish.

$$a=\frac{-102±\sqrt{10404+3584}}{2\times 4}$$

10404 ni 3584 ga qo'shish.

$$a=\frac{-102±\sqrt{13988}}{2\times 4}$$

13988 ning kvadrat ildizini chiqarish.

$$a=\frac{-102±2\sqrt{3497}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$a=\frac{-102±2\sqrt{3497}}{8}$$

a=\frac{-102±2\sqrt{3497}}{8} tenglamasini yeching, bunda ± musbat. -102 ni 2\sqrt{3497} ga qo'shish.

$$a=\frac{2\sqrt{3497}-102}{8}$$

-102+2\sqrt{3497} ni 8 ga bo'lish.

$$a=\frac{\sqrt{3497}-51}{4}$$

a=\frac{-102±2\sqrt{3497}}{8} tenglamasini yeching, bunda ± manfiy. -102 dan 2\sqrt{3497} ni ayirish.

$$a=\frac{-2\sqrt{3497}-102}{8}$$

-102-2\sqrt{3497} ni 8 ga bo'lish.

$$a=\frac{-\sqrt{3497}-51}{4}$$

Tenglama yechildi.

$$a=\frac{\sqrt{3497}-51}{4} a=\frac{-\sqrt{3497}-51}{4}$$

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.

$$4a^{2}+102a-224=0$$

224 ni tenglamaning ikkala tarafiga qo'shish.

$$4a^{2}+102a-224-\left(-224\right)=-\left(-224\right)$$

O‘zidan -224 ayirilsa 0 qoladi.

$$4a^{2}+102a=-\left(-224\right)$$

0 dan -224 ni ayirish.

$$4a^{2}+102a=224$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4a^{2}+102a}{4}=\frac{224}{4}$$

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

$$a^{2}+\frac{102}{4}a=\frac{224}{4}$$

\frac{102}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a^{2}+\frac{51}{2}a=\frac{224}{4}$$

224 ni 4 ga bo'lish.

$$a^{2}+\frac{51}{2}a=56$$

\frac{51}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{51}{4} olish uchun. Keyin, \frac{51}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}+\frac{51}{2}a+\left(\frac{51}{4}\right)^{2}=56+\left(\frac{51}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{51}{4} kvadratini chiqarish.

$$a^{2}+\frac{51}{2}a+\frac{2601}{16}=56+\frac{2601}{16}$$

56 ni \frac{2601}{16} ga qo'shish.

$$a^{2}+\frac{51}{2}a+\frac{2601}{16}=\frac{3497}{16}$$

a^{2}+\frac{51}{2}a+\frac{2601}{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.

$$\left(a+\frac{51}{4}\right)^{2}=\frac{3497}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a+\frac{51}{4}\right)^{2}}=\sqrt{\frac{3497}{16}}$$

Qisqartirish.

$$a+\frac{51}{4}=\frac{\sqrt{3497}}{4} a+\frac{51}{4}=-\frac{\sqrt{3497}}{4}$$

Tenglamaning ikkala tarafidan \frac{51}{4} ni ayirish.

$$a=\frac{\sqrt{3497}-51}{4} a=\frac{-\sqrt{3497}-51}{4}$$