Solve x^2-7x-15

$${ x }^{ 2 } -7x-15$$

Factor

$$\left(x-\frac{7-\sqrt{109}}{2}\right)\left(x-\frac{\sqrt{109}+7}{2}\right)$$

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.

$$x^{2}-7x-15=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-15\right)}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-\left(-7\right)±\sqrt{49+60}}{2}$$

Add 49 to 60.

$$x=\frac{-\left(-7\right)±\sqrt{109}}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{109}}{2}$$

Now solve the equation x=\frac{7±\sqrt{109}}{2} when ± is plus. Add 7 to \sqrt{109}.

$$x=\frac{\sqrt{109}+7}{2}$$

Now solve the equation x=\frac{7±\sqrt{109}}{2} when ± is minus. Subtract \sqrt{109} from 7.

$$x=\frac{7-\sqrt{109}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{109}}{2} for x_{1} and \frac{7-\sqrt{109}}{2} for x_{2}.

$$x^{2}-7x-15=\left(x-\frac{\sqrt{109}+7}{2}\right)\left(x-\frac{7-\sqrt{109}}{2}\right)$$

Evaluate

$$x^{2}-7x-15$$