2x^2-14x-54=0 eching

$$2 { x }^{ 2 } -14x-54=0$$

x uchun yechish

$$x = \frac{\sqrt{157} + 7}{2} \approx 9,764982043$$

$$x=\frac{7-\sqrt{157}}{2}\approx -2,764982043$$

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.

$$2x^{2}-14x-54=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -14 ni b va -54 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 2\left(-54\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-8\left(-54\right)}}{2\times 2}$$

-8 ni -54 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+432}}{2\times 2}$$

196 ni 432 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{628}}{2\times 2}$$

628 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2\sqrt{157}}{2\times 2}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2\sqrt{157}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{14±2\sqrt{157}}{4}$$

x=\frac{14±2\sqrt{157}}{4} tenglamasini yeching, bunda ± musbat. 14 ni 2\sqrt{157} ga qo'shish.

$$x=\frac{2\sqrt{157}+14}{4}$$

14+2\sqrt{157} ni 4 ga bo'lish.

$$x=\frac{\sqrt{157}+7}{2}$$

x=\frac{14±2\sqrt{157}}{4} tenglamasini yeching, bunda ± manfiy. 14 dan 2\sqrt{157} ni ayirish.

$$x=\frac{14-2\sqrt{157}}{4}$$

14-2\sqrt{157} ni 4 ga bo'lish.

$$x=\frac{7-\sqrt{157}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{157}+7}{2} x=\frac{7-\sqrt{157}}{2}$$

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.

$$2x^{2}-14x-54=0$$

54 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-14x-54-\left(-54\right)=-\left(-54\right)$$

O‘zidan -54 ayirilsa 0 qoladi.

$$2x^{2}-14x=-\left(-54\right)$$

0 dan -54 ni ayirish.

$$2x^{2}-14x=54$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-14x}{2}=\frac{54}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{14}{2}\right)x=\frac{54}{2}$$

-14 ni 2 ga bo'lish.

$$x^{2}-7x=\frac{54}{2}$$

54 ni 2 ga bo'lish.

$$x^{2}-7x=27$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=27+\left(-\frac{7}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=27+\frac{49}{4}$$

27 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{157}{4}$$

x^{2}-7x+\frac{49}{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.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{157}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{157}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{\sqrt{157}}{2} x-\frac{7}{2}=-\frac{\sqrt{157}}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{157}+7}{2} x=\frac{7-\sqrt{157}}{2}$$