5x^2+7x=2 eching

$$5 x ^ { 2 } + 7 x = 2$$

x uchun yechish

$$x=\frac{\sqrt{89}-7}{10}\approx 0,243398113$$

$$x=\frac{-\sqrt{89}-7}{10}\approx -1,643398113$$

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.

$$5x^{2}+7x=2$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$5x^{2}+7x-2=2-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$5x^{2}+7x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 7 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 5\left(-2\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-20\left(-2\right)}}{2\times 5}$$

-20 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+40}}{2\times 5}$$

49 ni 40 ga qo'shish.

$$x=\frac{-7±\sqrt{89}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{89}}{10}$$

x=\frac{-7±\sqrt{89}}{10} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{89} ga qo'shish.

$$x=\frac{\sqrt{89}-7}{10}$$

x=\frac{-7±\sqrt{89}}{10} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{89} ni ayirish.

$$x=\frac{-\sqrt{89}-7}{10}$$

Tenglama yechildi.

$$x=\frac{\sqrt{89}-7}{10} x=\frac{-\sqrt{89}-7}{10}$$

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.

$$5x^{2}+7x=2$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+7x}{5}=\frac{2}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{5}x=\frac{2}{5}$$

\frac{7}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{10} olish uchun. Keyin, \frac{7}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{5}x+\left(\frac{7}{10}\right)^{2}=\frac{2}{5}+\left(\frac{7}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{10} kvadratini chiqarish.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{2}{5}+\frac{49}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{5} ni \frac{49}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{89}{100}$$

x^{2}+\frac{7}{5}x+\frac{49}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{10}\right)^{2}=\frac{89}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{10}\right)^{2}}=\sqrt{\frac{89}{100}}$$

Qisqartirish.

$$x+\frac{7}{10}=\frac{\sqrt{89}}{10} x+\frac{7}{10}=-\frac{\sqrt{89}}{10}$$

Tenglamaning ikkala tarafidan \frac{7}{10} ni ayirish.

$$x=\frac{\sqrt{89}-7}{10} x=\frac{-\sqrt{89}-7}{10}$$