x^2-6x-11=4 eching

$${ x }^{ 2 } -6x-11=4$$

x uchun yechish

$$x=2\sqrt{6}+3\approx 7,898979486$$

$$x=3-2\sqrt{6}\approx -1,898979486$$

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.

$$x^{2}-6x-11=4$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$x^{2}-6x-11-4=4-4$$

O‘zidan 4 ayirilsa 0 qoladi.

$$x^{2}-6x-11-4=0$$

-11 dan 4 ni ayirish.

$$x^{2}-6x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -6 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-15\right)}}{2}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-15\right)}}{2}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+60}}{2}$$

36 ni 60 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{96}}{2}$$

96 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±4\sqrt{6}}{2}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±4\sqrt{6}}{2}$$

x=\frac{6±4\sqrt{6}}{2} tenglamasini yeching, bunda ± musbat. 6 ni 4\sqrt{6} ga qo'shish.

$$x=\frac{4\sqrt{6}+6}{2}$$

6+4\sqrt{6} ni 2 ga bo'lish.

$$x=2\sqrt{6}+3$$

x=\frac{6±4\sqrt{6}}{2} tenglamasini yeching, bunda ± manfiy. 6 dan 4\sqrt{6} ni ayirish.

$$x=\frac{6-4\sqrt{6}}{2}$$

6-4\sqrt{6} ni 2 ga bo'lish.

$$x=3-2\sqrt{6}$$

Tenglama yechildi.

$$x=2\sqrt{6}+3 x=3-2\sqrt{6}$$

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.

$$x^{2}-6x-11=4$$

11 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}-6x-11-\left(-11\right)=4-\left(-11\right)$$

O‘zidan -11 ayirilsa 0 qoladi.

$$x^{2}-6x=4-\left(-11\right)$$

4 dan -11 ni ayirish.

$$x^{2}-6x=15$$

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

$$x^{2}-6x+\left(-3\right)^{2}=15+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=15+9$$

15 ni 9 ga qo'shish.

$$x^{2}-6x+9=24$$

x^{2}-6x+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-3\right)^{2}=24$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{24}$$

Qisqartirish.

$$x-3=2\sqrt{6} x-3=-2\sqrt{6}$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{6}+3 x=3-2\sqrt{6}$$