2x^2+3x=7 eching

$$2 x ^ { 2 } + 3 x = 7$$

x uchun yechish

$$x = \frac{\sqrt{65} - 3}{4} \approx 1,265564437$$

$$x=\frac{-\sqrt{65}-3}{4}\approx -2,765564437$$

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.

$$2x^{2}+3x=7$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$2x^{2}+3x-7=7-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$2x^{2}+3x-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 3 ni b va -7 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-7\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-8\left(-7\right)}}{2\times 2}$$

-8 ni -7 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+56}}{2\times 2}$$

9 ni 56 ga qo'shish.

$$x=\frac{-3±\sqrt{65}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{65}}{4}$$

x=\frac{-3±\sqrt{65}}{4} tenglamasini yeching, bunda ± musbat. -3 ni \sqrt{65} ga qo'shish.

$$x=\frac{\sqrt{65}-3}{4}$$

x=\frac{-3±\sqrt{65}}{4} tenglamasini yeching, bunda ± manfiy. -3 dan \sqrt{65} ni ayirish.

$$x=\frac{-\sqrt{65}-3}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{65}-3}{4} x=\frac{-\sqrt{65}-3}{4}$$

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.

$$2x^{2}+3x=7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+3x}{2}=\frac{7}{2}$$

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

$$x^{2}+\frac{3}{2}x=\frac{7}{2}$$

\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{4} olish uchun. Keyin, \frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{7}{2}+\left(\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{4} kvadratini chiqarish.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{7}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{65}{16}$$

x^{2}+\frac{3}{2}x+\frac{9}{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.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{65}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{65}{16}}$$

Qisqartirish.

$$x+\frac{3}{4}=\frac{\sqrt{65}}{4} x+\frac{3}{4}=-\frac{\sqrt{65}}{4}$$

Tenglamaning ikkala tarafidan \frac{3}{4} ni ayirish.

$$x=\frac{\sqrt{65}-3}{4} x=\frac{-\sqrt{65}-3}{4}$$