Solve -4x^2-28x+49geq0

$$-4 { x }^{ 2 } -28x+49 \geq 0$$

Solve for x

$$x\in \begin{bmatrix}\frac{-7\sqrt{2}-7}{2},\frac{7\sqrt{2}-7}{2}\end{bmatrix}$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -4x^{2}-28x+49 positive. Since -1 is negative, the inequality direction is changed.

$$4x^{2}+28x-49\leq 0$$

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.

$$4x^{2}+28x-49=0$$

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, 28 for b, and -49 for c in the quadratic formula.

$$x=\frac{-28±\sqrt{28^{2}-4\times 4\left(-49\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{-28±28\sqrt{2}}{8}$$

Solve the equation x=\frac{-28±28\sqrt{2}}{8} when ± is plus and when ± is minus.

$$x=\frac{7\sqrt{2}-7}{2} x=\frac{-7\sqrt{2}-7}{2}$$

Rewrite the inequality by using the obtained solutions.

$$4\left(x-\frac{7\sqrt{2}-7}{2}\right)\left(x-\frac{-7\sqrt{2}-7}{2}\right)\leq 0$$

For the product to be ≤0, one of the values x-\frac{7\sqrt{2}-7}{2} and x-\frac{-7\sqrt{2}-7}{2} has to be ≥0 and the other has to be ≤0. Consider the case when x-\frac{7\sqrt{2}-7}{2}\geq 0 and x-\frac{-7\sqrt{2}-7}{2}\leq 0.

$$x-\frac{7\sqrt{2}-7}{2}\geq 0 x-\frac{-7\sqrt{2}-7}{2}\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-\frac{7\sqrt{2}-7}{2}\leq 0 and x-\frac{-7\sqrt{2}-7}{2}\geq 0.

$$x-\frac{-7\sqrt{2}-7}{2}\geq 0 x-\frac{7\sqrt{2}-7}{2}\leq 0$$

The solution satisfying both inequalities is x\in \left[\frac{-7\sqrt{2}-7}{2},\frac{7\sqrt{2}-7}{2}\right].

$$x\in \begin{bmatrix}\frac{-7\sqrt{2}-7}{2},\frac{7\sqrt{2}-7}{2}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}\frac{-7\sqrt{2}-7}{2},\frac{7\sqrt{2}-7}{2}\end{bmatrix}$$