x^2+7x=10 eching

$$x ^ { 2 } + 7 x = 10$$

x uchun yechish

$$x = \frac{\sqrt{89} - 7}{2} \approx 1,216990566$$

$$x=\frac{-\sqrt{89}-7}{2}\approx -8,216990566$$

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.

$$x^{2}+7x=10$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$x^{2}+7x-10=10-10$$

O‘zidan 10 ayirilsa 0 qoladi.

$$x^{2}+7x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 7 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-10\right)}}{2}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\left(-10\right)}}{2}$$

-4 ni -10 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+40}}{2}$$

49 ni 40 ga qo'shish.

$$x=\frac{-7±\sqrt{89}}{2}$$

x=\frac{-7±\sqrt{89}}{2} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{89} ga qo'shish.

$$x=\frac{\sqrt{89}-7}{2}$$

x=\frac{-7±\sqrt{89}}{2} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{89} ni ayirish.

$$x=\frac{-\sqrt{89}-7}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{89}-7}{2} x=\frac{-\sqrt{89}-7}{2}$$

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.

$$x^{2}+7x=10$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{2} kvadratini chiqarish.

$$x^{2}+7x+\frac{49}{4}=10+\frac{49}{4}$$

10 ni \frac{49}{4} ga qo'shish.

$$x^{2}+7x+\frac{49}{4}=\frac{89}{4}$$

x^{2}+7x+\frac{49}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{89}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{89}{4}}$$

Qisqartirish.

$$x+\frac{7}{2}=\frac{\sqrt{89}}{2} x+\frac{7}{2}=-\frac{\sqrt{89}}{2}$$

Tenglamaning ikkala tarafidan \frac{7}{2} ni ayirish.

$$x=\frac{\sqrt{89}-7}{2} x=\frac{-\sqrt{89}-7}{2}$$