Solve 5x^2+48x-91

$$5 { x }^{ 2 } +48x-91$$

Factor

$$5\left(x-\frac{-\sqrt{1031}-24}{5}\right)\left(x-\frac{\sqrt{1031}-24}{5}\right)$$

Steps Using the Quadratic Formula

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.

$$5x^{2}+48x-91=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-48±\sqrt{48^{2}-4\times 5\left(-91\right)}}{2\times 5}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 5\left(-91\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-48±\sqrt{2304-20\left(-91\right)}}{2\times 5}$$

Multiply -20 times -91.

$$x=\frac{-48±\sqrt{2304+1820}}{2\times 5}$$

Add 2304 to 1820.

$$x=\frac{-48±\sqrt{4124}}{2\times 5}$$

Take the square root of 4124.

$$x=\frac{-48±2\sqrt{1031}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-48±2\sqrt{1031}}{10}$$

Now solve the equation x=\frac{-48±2\sqrt{1031}}{10} when ± is plus. Add -48 to 2\sqrt{1031}.

$$x=\frac{2\sqrt{1031}-48}{10}$$

Divide -48+2\sqrt{1031} by 10.

$$x=\frac{\sqrt{1031}-24}{5}$$

Now solve the equation x=\frac{-48±2\sqrt{1031}}{10} when ± is minus. Subtract 2\sqrt{1031} from -48.

$$x=\frac{-2\sqrt{1031}-48}{10}$$

Divide -48-2\sqrt{1031} by 10.

$$x=\frac{-\sqrt{1031}-24}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-24+\sqrt{1031}}{5} for x_{1} and \frac{-24-\sqrt{1031}}{5} for x_{2}.

$$5x^{2}+48x-91=5\left(x-\frac{\sqrt{1031}-24}{5}\right)\left(x-\frac{-\sqrt{1031}-24}{5}\right)$$

Evaluate

$$5x^{2}+48x-91$$