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

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

x uchun yechish

$$x = \frac{\sqrt{149} + 3}{14} \approx 1,086182544$$

$$x=\frac{3-\sqrt{149}}{14}\approx -0,657611115$$

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -3 ni b va -5 ni c bilan almashtiring.

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

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 7\left(-5\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-28\left(-5\right)}}{2\times 7}$$

-28 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+140}}{2\times 7}$$

9 ni 140 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{149}}{2\times 7}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{149}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{149}}{14}$$

x=\frac{3±\sqrt{149}}{14} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{149} ga qo'shish.

$$x=\frac{\sqrt{149}+3}{14}$$

x=\frac{3±\sqrt{149}}{14} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{149} ni ayirish.

$$x=\frac{3-\sqrt{149}}{14}$$

Tenglama yechildi.

$$x=\frac{\sqrt{149}+3}{14} x=\frac{3-\sqrt{149}}{14}$$

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.

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

5 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -5 ayirilsa 0 qoladi.

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

0 dan -5 ni ayirish.

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

Ikki tarafini 7 ga bo‘ling.

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

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

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

-\frac{3}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{14} olish uchun. Keyin, -\frac{3}{14} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{5}{7}+\left(-\frac{3}{14}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{14} kvadratini chiqarish.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{5}{7}+\frac{9}{196}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{7} ni \frac{9}{196} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{149}{196}$$

x^{2}-\frac{3}{7}x+\frac{9}{196} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{149}{196}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{149}{196}}$$

Qisqartirish.

$$x-\frac{3}{14}=\frac{\sqrt{149}}{14} x-\frac{3}{14}=-\frac{\sqrt{149}}{14}$$

\frac{3}{14} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{149}+3}{14} x=\frac{3-\sqrt{149}}{14}$$