5x^2-7x-3=0 eching

$$5 x ^ { 2 } - 7 x - 3 = 0$$

x uchun yechish

$$x = \frac{\sqrt{109} + 7}{10} \approx 1,744030651$$

$$x=\frac{7-\sqrt{109}}{10}\approx -0,344030651$$

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.

$$5x^{2}-7x-3=0$$

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 -3 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 5\left(-3\right)}}{2\times 5}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 5\left(-3\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-20\left(-3\right)}}{2\times 5}$$

-20 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+60}}{2\times 5}$$

49 ni 60 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{109}}{2\times 5}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{109}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{109}}{10}$$

x=\frac{7±\sqrt{109}}{10} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{109} ga qo'shish.

$$x=\frac{\sqrt{109}+7}{10}$$

x=\frac{7±\sqrt{109}}{10} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{109} ni ayirish.

$$x=\frac{7-\sqrt{109}}{10}$$

Tenglama yechildi.

$$x=\frac{\sqrt{109}+7}{10} x=\frac{7-\sqrt{109}}{10}$$

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.

$$5x^{2}-7x-3=0$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$5x^{2}-7x-3-\left(-3\right)=-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$5x^{2}-7x=-\left(-3\right)$$

0 dan -3 ni ayirish.

$$5x^{2}-7x=3$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-7x}{5}=\frac{3}{5}$$

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

$$x^{2}-\frac{7}{5}x=\frac{3}{5}$$

-\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.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=\frac{3}{5}+\left(-\frac{7}{10}\right)^{2}$$

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

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{3}{5}+\frac{49}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{5} ni \frac{49}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{109}{100}$$

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.

$$\left(x-\frac{7}{10}\right)^{2}=\frac{109}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{\frac{109}{100}}$$

Qisqartirish.

$$x-\frac{7}{10}=\frac{\sqrt{109}}{10} x-\frac{7}{10}=-\frac{\sqrt{109}}{10}$$

\frac{7}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{109}+7}{10} x=\frac{7-\sqrt{109}}{10}$$