4x^2-4x-16=0 eching

$$4 { x }^{ 2 } -4x-16=0$$

x uchun yechish

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

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

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.

$$4x^{2}-4x-16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -4 ni b va -16 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-16\right)}}{2\times 4}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-16\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-16\right)}}{2\times 4}$$

-16 ni -16 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+256}}{2\times 4}$$

16 ni 256 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{272}}{2\times 4}$$

272 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±4\sqrt{17}}{2\times 4}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±4\sqrt{17}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{4±4\sqrt{17}}{8}$$

x=\frac{4±4\sqrt{17}}{8} tenglamasini yeching, bunda ± musbat. 4 ni 4\sqrt{17} ga qo'shish.

$$x=\frac{4\sqrt{17}+4}{8}$$

4+4\sqrt{17} ni 8 ga bo'lish.

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

x=\frac{4±4\sqrt{17}}{8} tenglamasini yeching, bunda ± manfiy. 4 dan 4\sqrt{17} ni ayirish.

$$x=\frac{4-4\sqrt{17}}{8}$$

4-4\sqrt{17} ni 8 ga bo'lish.

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

Tenglama yechildi.

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

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.

$$4x^{2}-4x-16=0$$

16 ni tenglamaning ikkala tarafiga qo'shish.

$$4x^{2}-4x-16-\left(-16\right)=-\left(-16\right)$$

O‘zidan -16 ayirilsa 0 qoladi.

$$4x^{2}-4x=-\left(-16\right)$$

0 dan -16 ni ayirish.

$$4x^{2}-4x=16$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-4x}{4}=\frac{16}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{16}{4}$$

-4 ni 4 ga bo'lish.

$$x^{2}-x=\frac{16}{4}$$

16 ni 4 ga bo'lish.

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

-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.

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

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

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