x^2+4x-11<-36x-x^2 eching

$$x ^ { 2 } + 4 x - 11 < - 36 x - x ^ { 2 }$$

x uchun yechish

$$x\in \left(-\frac{\sqrt{422}}{2}-10,\frac{\sqrt{422}}{2}-10\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

36x ni ikki tarafga qo’shing.

$$x^{2}+4x-11+36x<-x^{2}$$

40x ni olish uchun 4x va 36x ni birlashtirish.

$$x^{2}+40x-11<-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}+40x-11+x^{2}<0$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$2x^{2}+40x-11<0$$

Tengsizlikni yechish uchun chap tomon faktorini hisoblang. Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$2x^{2}+40x-11=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 2 ni, b uchun 40 ni va c uchun -11 ni ayiring.

$$x=\frac{-40±\sqrt{40^{2}-4\times 2\left(-11\right)}}{2\times 2}$$

Hisoblarni amalga oshiring.

$$x=\frac{-40±2\sqrt{422}}{4}$$

x=\frac{-40±2\sqrt{422}}{4} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=\frac{\sqrt{422}}{2}-10 x=-\frac{\sqrt{422}}{2}-10$$

Yechimlardan foydalanib tengsizlikni qaytadan yozing.

$$2\left(x-\left(\frac{\sqrt{422}}{2}-10\right)\right)\left(x-\left(-\frac{\sqrt{422}}{2}-10\right)\right)<0$$

Koʻpaytma manfiy boʻlishi uchun x-\left(\frac{\sqrt{422}}{2}-10\right) va x-\left(-\frac{\sqrt{422}}{2}-10\right) qarama-qarshi belgilar boʻlishi kerak. x-\left(\frac{\sqrt{422}}{2}-10\right) musbat, x-\left(-\frac{\sqrt{422}}{2}-10\right) manfiy boʻlganda, yechimni toping.

$$x-\left(\frac{\sqrt{422}}{2}-10\right)>0 x-\left(-\frac{\sqrt{422}}{2}-10\right)<0$$

Bu har qanday x uchun xato.

$$x\in \emptyset$$

x-\left(-\frac{\sqrt{422}}{2}-10\right) musbat, x-\left(\frac{\sqrt{422}}{2}-10\right) manfiy boʻlganda, yechimni toping.

$$x-\left(-\frac{\sqrt{422}}{2}-10\right)>0 x-\left(\frac{\sqrt{422}}{2}-10\right)<0$$

Ikkala tengsizlikning mos yechimi – x\in \left(-\frac{\sqrt{422}}{2}-10,\frac{\sqrt{422}}{2}-10\right).

$$x\in \left(-\frac{\sqrt{422}}{2}-10,\frac{\sqrt{422}}{2}-10\right)$$

Oxirgi yechim olingan yechimlarning birlashmasidir.

$$x\in \left(-\frac{\sqrt{422}}{2}-10,\frac{\sqrt{422}}{2}-10\right)$$