3x^2+5x-10=0 eching

$$3 x ^ { 2 } + 5 x - 10 = 0$$

x uchun yechish

$$x = \frac{\sqrt{145} - 5}{6} \approx 1,173599096$$

$$x=\frac{-\sqrt{145}-5}{6}\approx -2,840265763$$

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.

$$3x^{2}+5x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 5 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-10\right)}}{2\times 3}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-10\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-12\left(-10\right)}}{2\times 3}$$

-12 ni -10 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+120}}{2\times 3}$$

25 ni 120 ga qo'shish.

$$x=\frac{-5±\sqrt{145}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{145}}{6}$$

x=\frac{-5±\sqrt{145}}{6} tenglamasini yeching, bunda ± musbat. -5 ni \sqrt{145} ga qo'shish.

$$x=\frac{\sqrt{145}-5}{6}$$

x=\frac{-5±\sqrt{145}}{6} tenglamasini yeching, bunda ± manfiy. -5 dan \sqrt{145} ni ayirish.

$$x=\frac{-\sqrt{145}-5}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{145}-5}{6} x=\frac{-\sqrt{145}-5}{6}$$

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.

$$3x^{2}+5x-10=0$$

10 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -10 ayirilsa 0 qoladi.

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

0 dan -10 ni ayirish.

$$3x^{2}+5x=10$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+5x}{3}=\frac{10}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{5}{3}x=\frac{10}{3}$$

\frac{5}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{6} olish uchun. Keyin, \frac{5}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{10}{3}+\left(\frac{5}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{6} kvadratini chiqarish.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{10}{3}+\frac{25}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10}{3} ni \frac{25}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{145}{36}$$

x^{2}+\frac{5}{3}x+\frac{25}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{145}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{145}{36}}$$

Qisqartirish.

$$x+\frac{5}{6}=\frac{\sqrt{145}}{6} x+\frac{5}{6}=-\frac{\sqrt{145}}{6}$$

Tenglamaning ikkala tarafidan \frac{5}{6} ni ayirish.

$$x=\frac{\sqrt{145}-5}{6} x=\frac{-\sqrt{145}-5}{6}$$