Solve 4x(x-4)+7geq0

$$4 x ( x - 4 ) + 7 \geq 0$$

Solve for x

$$x\in (-\infty,\frac{1}{2}]\cup [\frac{7}{2},\infty)$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x-4.

$$4x^{2}-16x+7\geq 0$$

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.

$$4x^{2}-16x+7=0$$

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, -16 for b, and 7 for c in the quadratic formula.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Do the calculations.

$$x=\frac{16±12}{8}$$

Solve the equation x=\frac{16±12}{8} when ± is plus and when ± is minus.

$$x=\frac{7}{2} x=\frac{1}{2}$$

Rewrite the inequality by using the obtained solutions.

$$4\left(x-\frac{7}{2}\right)\left(x-\frac{1}{2}\right)\geq 0$$

For the product to be ≥0, x-\frac{7}{2} and x-\frac{1}{2} have to be both ≤0 or both ≥0. Consider the case when x-\frac{7}{2} and x-\frac{1}{2} are both ≤0.

$$x-\frac{7}{2}\leq 0 x-\frac{1}{2}\leq 0$$

The solution satisfying both inequalities is x\leq \frac{1}{2}.

$$x\leq \frac{1}{2}$$

Consider the case when x-\frac{7}{2} and x-\frac{1}{2} are both ≥0.

$$x-\frac{1}{2}\geq 0 x-\frac{7}{2}\geq 0$$

The solution satisfying both inequalities is x\geq \frac{7}{2}.

$$x\geq \frac{7}{2}$$

The final solution is the union of the obtained solutions.

$$x\leq \frac{1}{2}\text{; }x\geq \frac{7}{2}$$