-6x^2-18x+20geq0 eching

$$- 6 x ^ { 2 } - 18 x + 20 \geq 0$$

x uchun yechish

$$x\in \begin{bmatrix}-\frac{\sqrt{201}}{6}-\frac{3}{2},\frac{\sqrt{201}}{6}-\frac{3}{2}\end{bmatrix}$$

Kvadrat tenglamalaridan foydalanish qoidalari

-6x^{2}-18x+20 musbatida eng katta quvvatni koeffitsientini aniqlash uchun tengsizlikni -1 ga koʻpaytiring. -1 manfiy boʻlgani uchun tengsizlikning yo‘nalishi o‘zgaradi.

$$6x^{2}+18x-20\leq 0$$

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.

$$6x^{2}+18x-20=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 6 ni, b uchun 18 ni va c uchun -20 ni ayiring.

$$x=\frac{-18±\sqrt{18^{2}-4\times 6\left(-20\right)}}{2\times 6}$$

Hisoblarni amalga oshiring.

$$x=\frac{-18±2\sqrt{201}}{12}$$

x=\frac{-18±2\sqrt{201}}{12} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=\frac{\sqrt{201}}{6}-\frac{3}{2} x=-\frac{\sqrt{201}}{6}-\frac{3}{2}$$

Yechimlardan foydalanib tengsizlikni qaytadan yozing.

$$6\left(x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)\left(x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\right)\leq 0$$

Koʻpaytma ≤0 boʻlishi uchun qiymatlardan biri x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right) va x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right) ≥0 va boshqasi ≤0 boʻlishi kerak. x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\geq 0 va x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\leq 0 boʻlgandagi holatni koʻrib chiqing.

$$x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\geq 0 x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\leq 0$$

Bu har qanday x uchun xato.

$$x\in \emptyset$$

x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\leq 0 va x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\geq 0 boʻlgandagi holatni koʻrib chiqing.

$$x-\left(-\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\geq 0 x-\left(\frac{\sqrt{201}}{6}-\frac{3}{2}\right)\leq 0$$

Ikkala tengsizlikning mos yechimi – x\in \left[-\frac{\sqrt{201}}{6}-\frac{3}{2},\frac{\sqrt{201}}{6}-\frac{3}{2}\right].

$$x\in \begin{bmatrix}-\frac{\sqrt{201}}{6}-\frac{3}{2},\frac{\sqrt{201}}{6}-\frac{3}{2}\end{bmatrix}$$

Oxirgi yechim olingan yechimlarning birlashmasidir.

$$x\in \begin{bmatrix}-\frac{\sqrt{201}}{6}-\frac{3}{2},\frac{\sqrt{201}}{6}-\frac{3}{2}\end{bmatrix}$$