Solve x^2+23x-45

$${ x }^{ 2 } +23x-45$$

Factor

$$\left(x-\frac{-\sqrt{709}-23}{2}\right)\left(x-\frac{\sqrt{709}-23}{2}\right)$$

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.

$$x^{2}+23x-45=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\left(-45\right)}}{2}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$x=\frac{-23±\sqrt{529+180}}{2}$$

Add 529 to 180.

$$x=\frac{-23±\sqrt{709}}{2}$$

Now solve the equation x=\frac{-23±\sqrt{709}}{2} when ± is plus. Add -23 to \sqrt{709}.

$$x=\frac{\sqrt{709}-23}{2}$$

Now solve the equation x=\frac{-23±\sqrt{709}}{2} when ± is minus. Subtract \sqrt{709} from -23.

$$x=\frac{-\sqrt{709}-23}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-23+\sqrt{709}}{2} for x_{1} and \frac{-23-\sqrt{709}}{2} for x_{2}.

$$x^{2}+23x-45=\left(x-\frac{\sqrt{709}-23}{2}\right)\left(x-\frac{-\sqrt{709}-23}{2}\right)$$

Evaluate

$$x^{2}+23x-45$$