x+x^2-1=3 eching

$$x + x ^ { 2 } - 1 = 3$$

x uchun yechish

$$x = \frac{\sqrt{17} - 1}{2} \approx 1,561552813$$

$$x=\frac{-\sqrt{17}-1}{2}\approx -2,561552813$$

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.

$$x^{2}+x-1=3$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$x^{2}+x-1-3=3-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$x^{2}+x-1-3=0$$

-1 dan 3 ni ayirish.

$$x^{2}+x-4=0$$

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 -4 ni c bilan almashtiring.

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

1 kvadratini chiqarish.

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

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+16}}{2}$$

1 ni 16 ga qo'shish.

$$x=\frac{-1±\sqrt{17}}{2}$$

x=\frac{-1±\sqrt{17}}{2} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}-1}{2}$$

x=\frac{-1±\sqrt{17}}{2} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{17} ni ayirish.

$$x=\frac{-\sqrt{17}-1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{17}-1}{2} x=\frac{-\sqrt{17}-1}{2}$$

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.

$$x^{2}+x-1=3$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+x-1-\left(-1\right)=3-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$x^{2}+x=3-\left(-1\right)$$

3 dan -1 ni ayirish.

$$x^{2}+x=4$$

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.

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

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

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

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

$$x^{2}+x+\frac{1}{4}=\frac{17}{4}$$

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.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{17}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{17}}{2} x+\frac{1}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}-1}{2} x=\frac{-\sqrt{17}-1}{2}$$