x+x^2=5 eching

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

x uchun yechish

$$x = \frac{\sqrt{21} - 1}{2} \approx 1,791287847$$

$$x=\frac{-\sqrt{21}-1}{2}\approx -2,791287847$$

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.

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

Tenglamaning ikkala tarafidan 5 ni ayirish.

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

O‘zidan 5 ayirilsa 0 qoladi.

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

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

$$x=\frac{-1±\sqrt{1^{2}-4\left(-5\right)}}{2}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\left(-5\right)}}{2}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+20}}{2}$$

1 ni 20 ga qo'shish.

$$x=\frac{-1±\sqrt{21}}{2}$$

x=\frac{-1±\sqrt{21}}{2} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{21} ga qo'shish.

$$x=\frac{\sqrt{21}-1}{2}$$

x=\frac{-1±\sqrt{21}}{2} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{21} ni ayirish.

$$x=\frac{-\sqrt{21}-1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{21}-1}{2} x=\frac{-\sqrt{21}-1}{2}$$

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.

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

1 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{2} olish uchun. Keyin, \frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=5+\left(\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=5+\frac{1}{4}$$

5 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{21}{4}$$

x^{2}+x+\frac{1}{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.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{21}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{21}}{2} x+\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{\sqrt{21}-1}{2} x=\frac{-\sqrt{21}-1}{2}$$