2a^2-21a+48=0 eching

$$2 a ^ { 2 } - 21 a + 48 = 0$$

a uchun yechish

$$a = \frac{\sqrt{57} + 21}{4} \approx 7,137458609$$

$$a = \frac{21 - \sqrt{57}}{4} \approx 3,362541391$$

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.

$$2a^{2}-21a+48=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -21 ni b va 48 ni c bilan almashtiring.

$$a=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 2\times 48}}{2\times 2}$$

-21 kvadratini chiqarish.

$$a=\frac{-\left(-21\right)±\sqrt{441-4\times 2\times 48}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$a=\frac{-\left(-21\right)±\sqrt{441-8\times 48}}{2\times 2}$$

-8 ni 48 marotabaga ko'paytirish.

$$a=\frac{-\left(-21\right)±\sqrt{441-384}}{2\times 2}$$

441 ni -384 ga qo'shish.

$$a=\frac{-\left(-21\right)±\sqrt{57}}{2\times 2}$$

-21 ning teskarisi 21 ga teng.

$$a=\frac{21±\sqrt{57}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$a=\frac{21±\sqrt{57}}{4}$$

a=\frac{21±\sqrt{57}}{4} tenglamasini yeching, bunda ± musbat. 21 ni \sqrt{57} ga qo'shish.

$$a=\frac{\sqrt{57}+21}{4}$$

a=\frac{21±\sqrt{57}}{4} tenglamasini yeching, bunda ± manfiy. 21 dan \sqrt{57} ni ayirish.

$$a=\frac{21-\sqrt{57}}{4}$$

Tenglama yechildi.

$$a=\frac{\sqrt{57}+21}{4} a=\frac{21-\sqrt{57}}{4}$$

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.

$$2a^{2}-21a+48=0$$

Tenglamaning ikkala tarafidan 48 ni ayirish.

$$2a^{2}-21a+48-48=-48$$

O‘zidan 48 ayirilsa 0 qoladi.

$$2a^{2}-21a=-48$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2a^{2}-21a}{2}=-\frac{48}{2}$$

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

$$a^{2}-\frac{21}{2}a=-\frac{48}{2}$$

-48 ni 2 ga bo'lish.

$$a^{2}-\frac{21}{2}a=-24$$

-\frac{21}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{21}{4} olish uchun. Keyin, -\frac{21}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}-\frac{21}{2}a+\left(-\frac{21}{4}\right)^{2}=-24+\left(-\frac{21}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{21}{4} kvadratini chiqarish.

$$a^{2}-\frac{21}{2}a+\frac{441}{16}=-24+\frac{441}{16}$$

-24 ni \frac{441}{16} ga qo'shish.

$$a^{2}-\frac{21}{2}a+\frac{441}{16}=\frac{57}{16}$$

a^{2}-\frac{21}{2}a+\frac{441}{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.

$$\left(a-\frac{21}{4}\right)^{2}=\frac{57}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a-\frac{21}{4}\right)^{2}}=\sqrt{\frac{57}{16}}$$

Qisqartirish.

$$a-\frac{21}{4}=\frac{\sqrt{57}}{4} a-\frac{21}{4}=-\frac{\sqrt{57}}{4}$$

\frac{21}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$a=\frac{\sqrt{57}+21}{4} a=\frac{21-\sqrt{57}}{4}$$