Solve x^2-x-72leq0

$$x ^ { 2 } - x - 72 \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-8,9\end{bmatrix}$$

Steps Using the Quadratic Formula

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.

$$x^{2}-x-72=0$$

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 1 for a, -1 for b, and -72 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\left(-72\right)}}{2}$$

Do the calculations.

$$x=\frac{1±17}{2}$$

Solve the equation x=\frac{1±17}{2} when ± is plus and when ± is minus.

$$x=9 x=-8$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-9\right)\left(x+8\right)\leq 0$$

For the product to be ≤0, one of the values x-9 and x+8 has to be ≥0 and the other has to be ≤0. Consider the case when x-9\geq 0 and x+8\leq 0.

$$x-9\geq 0 x+8\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-9\leq 0 and x+8\geq 0.

$$x+8\geq 0 x-9\leq 0$$

The solution satisfying both inequalities is x\in \left[-8,9\right].

$$x\in \begin{bmatrix}-8,9\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-8,9\end{bmatrix}$$