x^2-14x-4=0 eching

$$x ^ { 2 } - 14 x - 4 = 0$$

x uchun yechish

$$x=\sqrt{53}+7\approx 14,280109889$$

$$x=7-\sqrt{53}\approx -0,280109889$$

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.

$$x^{2}-14x-4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -14 ni b va -4 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-4\right)}}{2}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-4\right)}}{2}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+16}}{2}$$

196 ni 16 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{212}}{2}$$

212 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2\sqrt{53}}{2}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2\sqrt{53}}{2}$$

x=\frac{14±2\sqrt{53}}{2} tenglamasini yeching, bunda ± musbat. 14 ni 2\sqrt{53} ga qo'shish.

$$x=\frac{2\sqrt{53}+14}{2}$$

14+2\sqrt{53} ni 2 ga bo'lish.

$$x=\sqrt{53}+7$$

x=\frac{14±2\sqrt{53}}{2} tenglamasini yeching, bunda ± manfiy. 14 dan 2\sqrt{53} ni ayirish.

$$x=\frac{14-2\sqrt{53}}{2}$$

14-2\sqrt{53} ni 2 ga bo'lish.

$$x=7-\sqrt{53}$$

Tenglama yechildi.

$$x=\sqrt{53}+7 x=7-\sqrt{53}$$

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.

$$x^{2}-14x-4=0$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}-14x-4-\left(-4\right)=-\left(-4\right)$$

O‘zidan -4 ayirilsa 0 qoladi.

$$x^{2}-14x=-\left(-4\right)$$

0 dan -4 ni ayirish.

$$x^{2}-14x=4$$

-14 ni bo‘lish, x shartining koeffitsienti, 2 ga -7 olish uchun. Keyin, -7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-14x+\left(-7\right)^{2}=4+\left(-7\right)^{2}$$

-7 kvadratini chiqarish.

$$x^{2}-14x+49=4+49$$

4 ni 49 ga qo'shish.

$$x^{2}-14x+49=53$$

x^{2}-14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-7\right)^{2}=53$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{53}$$

Qisqartirish.

$$x-7=\sqrt{53} x-7=-\sqrt{53}$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{53}+7 x=7-\sqrt{53}$$