x^2+3x-9=27 eching

$${ x }^{ 2 } +3x-9=27$$

x uchun yechish

$$x = \frac{3 \sqrt{17} - 3}{2} \approx 4,684658438$$

$$x=\frac{-3\sqrt{17}-3}{2}\approx -7,684658438$$

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.

$$x^{2}+3x-9=27$$

Tenglamaning ikkala tarafidan 27 ni ayirish.

$$x^{2}+3x-9-27=27-27$$

O‘zidan 27 ayirilsa 0 qoladi.

$$x^{2}+3x-9-27=0$$

-9 dan 27 ni ayirish.

$$x^{2}+3x-36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 3 ni b va -36 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-36\right)}}{2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-36\right)}}{2}$$

-4 ni -36 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+144}}{2}$$

9 ni 144 ga qo'shish.

$$x=\frac{-3±\sqrt{153}}{2}$$

153 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±3\sqrt{17}}{2}$$

x=\frac{-3±3\sqrt{17}}{2} tenglamasini yeching, bunda ± musbat. -3 ni 3\sqrt{17} ga qo'shish.

$$x=\frac{3\sqrt{17}-3}{2}$$

x=\frac{-3±3\sqrt{17}}{2} tenglamasini yeching, bunda ± manfiy. -3 dan 3\sqrt{17} ni ayirish.

$$x=\frac{-3\sqrt{17}-3}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{17}-3}{2} x=\frac{-3\sqrt{17}-3}{2}$$

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.

$$x^{2}+3x-9=27$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+3x-9-\left(-9\right)=27-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$x^{2}+3x=27-\left(-9\right)$$

27 dan -9 ni ayirish.

$$x^{2}+3x=36$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=36+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=36+\frac{9}{4}$$

36 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=\frac{153}{4}$$

x^{2}+3x+\frac{9}{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.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{153}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{153}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{3\sqrt{17}}{2} x+\frac{3}{2}=-\frac{3\sqrt{17}}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=\frac{3\sqrt{17}-3}{2} x=\frac{-3\sqrt{17}-3}{2}$$