x^2+5x+10=15 eching

$$x ^ { 2 } + 5 x + 10 = 15$$

x uchun yechish

$$x=\frac{3\sqrt{5}-5}{2}\approx 0,854101966$$

$$x=\frac{-3\sqrt{5}-5}{2}\approx -5,854101966$$

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.

$$x^{2}+5x+10=15$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$x^{2}+5x+10-15=15-15$$

O‘zidan 15 ayirilsa 0 qoladi.

$$x^{2}+5x+10-15=0$$

10 dan 15 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 5 ni b va -5 ni c bilan almashtiring.

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

5 kvadratini chiqarish.

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

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+20}}{2}$$

25 ni 20 ga qo'shish.

$$x=\frac{-5±\sqrt{45}}{2}$$

45 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±3\sqrt{5}}{2}$$

x=\frac{-5±3\sqrt{5}}{2} tenglamasini yeching, bunda ± musbat. -5 ni 3\sqrt{5} ga qo'shish.

$$x=\frac{3\sqrt{5}-5}{2}$$

x=\frac{-5±3\sqrt{5}}{2} tenglamasini yeching, bunda ± manfiy. -5 dan 3\sqrt{5} ni ayirish.

$$x=\frac{-3\sqrt{5}-5}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{5}-5}{2} x=\frac{-3\sqrt{5}-5}{2}$$

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.

$$x^{2}+5x+10=15$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$x^{2}+5x+10-10=15-10$$

O‘zidan 10 ayirilsa 0 qoladi.

$$x^{2}+5x=15-10$$

15 dan 10 ni ayirish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$x^{2}+5x+\frac{25}{4}=5+\frac{25}{4}$$

5 ni \frac{25}{4} ga qo'shish.

$$x^{2}+5x+\frac{25}{4}=\frac{45}{4}$$

x^{2}+5x+\frac{25}{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.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{45}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Qisqartirish.

$$x+\frac{5}{2}=\frac{3\sqrt{5}}{2} x+\frac{5}{2}=-\frac{3\sqrt{5}}{2}$$

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$x=\frac{3\sqrt{5}-5}{2} x=\frac{-3\sqrt{5}-5}{2}$$