4x^2+14x-27=0 eching

$$4 x ^ { 2 } + 14 x - 27 = 0$$

x uchun yechish

$$x = \frac{\sqrt{157} - 7}{4} \approx 1,382491022$$

$$x=\frac{-\sqrt{157}-7}{4}\approx -4,882491022$$

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.

$$4x^{2}+14x-27=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 14 ni b va -27 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\times 4\left(-27\right)}}{2\times 4}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\times 4\left(-27\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196-16\left(-27\right)}}{2\times 4}$$

-16 ni -27 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+432}}{2\times 4}$$

196 ni 432 ga qo'shish.

$$x=\frac{-14±\sqrt{628}}{2\times 4}$$

628 ning kvadrat ildizini chiqarish.

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

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-14±2\sqrt{157}}{8}$$

x=\frac{-14±2\sqrt{157}}{8} tenglamasini yeching, bunda ± musbat. -14 ni 2\sqrt{157} ga qo'shish.

$$x=\frac{2\sqrt{157}-14}{8}$$

-14+2\sqrt{157} ni 8 ga bo'lish.

$$x=\frac{\sqrt{157}-7}{4}$$

x=\frac{-14±2\sqrt{157}}{8} tenglamasini yeching, bunda ± manfiy. -14 dan 2\sqrt{157} ni ayirish.

$$x=\frac{-2\sqrt{157}-14}{8}$$

-14-2\sqrt{157} ni 8 ga bo'lish.

$$x=\frac{-\sqrt{157}-7}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{157}-7}{4} x=\frac{-\sqrt{157}-7}{4}$$

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.

$$4x^{2}+14x-27=0$$

27 ni tenglamaning ikkala tarafiga qo'shish.

$$4x^{2}+14x-27-\left(-27\right)=-\left(-27\right)$$

O‘zidan -27 ayirilsa 0 qoladi.

$$4x^{2}+14x=-\left(-27\right)$$

0 dan -27 ni ayirish.

$$4x^{2}+14x=27$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+14x}{4}=\frac{27}{4}$$

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

$$x^{2}+\frac{14}{4}x=\frac{27}{4}$$

\frac{14}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{4} olish uchun. Keyin, \frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\frac{27}{4}+\left(\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{4} kvadratini chiqarish.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{27}{4}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{27}{4} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{157}{16}$$

x^{2}+\frac{7}{2}x+\frac{49}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{157}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{157}{16}}$$

Qisqartirish.

$$x+\frac{7}{4}=\frac{\sqrt{157}}{4} x+\frac{7}{4}=-\frac{\sqrt{157}}{4}$$

Tenglamaning ikkala tarafidan \frac{7}{4} ni ayirish.

$$x=\frac{\sqrt{157}-7}{4} x=\frac{-\sqrt{157}-7}{4}$$