Solve (2x)^2-12(x+1)geq0

$$( 2 x ) ^ { 2 } - 12 ( x + 1 ) \geq 0$$

Solve for x

$$x\in (-\infty,\frac{3-\sqrt{21}}{2}]\cup [\frac{\sqrt{21}+3}{2},\infty)$$

Steps Using the Quadratic Formula

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}-12\left(x+1\right)\geq 0$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-12\left(x+1\right)\geq 0$$

Use the distributive property to multiply -12 by x+1.

$$4x^{2}-12x-12\geq 0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$4x^{2}-12x-12=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 4 for a, -12 for b, and -12 for c in the quadratic formula.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-12\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{12±4\sqrt{21}}{8}$$

Solve the equation x=\frac{12±4\sqrt{21}}{8} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{21}+3}{2} x=\frac{3-\sqrt{21}}{2}$$

Rewrite the inequality by using the obtained solutions.

$$4\left(x-\frac{\sqrt{21}+3}{2}\right)\left(x-\frac{3-\sqrt{21}}{2}\right)\geq 0$$

For the product to be ≥0, x-\frac{\sqrt{21}+3}{2} and x-\frac{3-\sqrt{21}}{2} have to be both ≤0 or both ≥0. Consider the case when x-\frac{\sqrt{21}+3}{2} and x-\frac{3-\sqrt{21}}{2} are both ≤0.

$$x-\frac{\sqrt{21}+3}{2}\leq 0 x-\frac{3-\sqrt{21}}{2}\leq 0$$

The solution satisfying both inequalities is x\leq \frac{3-\sqrt{21}}{2}.

$$x\leq \frac{3-\sqrt{21}}{2}$$

Consider the case when x-\frac{\sqrt{21}+3}{2} and x-\frac{3-\sqrt{21}}{2} are both ≥0.

$$x-\frac{3-\sqrt{21}}{2}\geq 0 x-\frac{\sqrt{21}+3}{2}\geq 0$$

The solution satisfying both inequalities is x\geq \frac{\sqrt{21}+3}{2}.

$$x\geq \frac{\sqrt{21}+3}{2}$$

The final solution is the union of the obtained solutions.

$$x\leq \frac{3-\sqrt{21}}{2}\text{; }x\geq \frac{\sqrt{21}+3}{2}$$