Solve (x^2+13x+32)

$$( x ^ { 2 } + 13 x + 32 )$$

Factor

$$\left(x-\frac{-\sqrt{41}-13}{2}\right)\left(x-\frac{\sqrt{41}-13}{2}\right)$$

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.

$$x^{2}+13x+32=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 32}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 32}}{2}$$

Multiply -4 times 32.

$$x=\frac{-13±\sqrt{169-128}}{2}$$

Add 169 to -128.

$$x=\frac{-13±\sqrt{41}}{2}$$

Now solve the equation x=\frac{-13±\sqrt{41}}{2} when ± is plus. Add -13 to \sqrt{41}.

$$x=\frac{\sqrt{41}-13}{2}$$

Now solve the equation x=\frac{-13±\sqrt{41}}{2} when ± is minus. Subtract \sqrt{41} from -13.

$$x=\frac{-\sqrt{41}-13}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-13+\sqrt{41}}{2} for x_{1} and \frac{-13-\sqrt{41}}{2} for x_{2}.

$$x^{2}+13x+32=\left(x-\frac{\sqrt{41}-13}{2}\right)\left(x-\frac{-\sqrt{41}-13}{2}\right)$$

Evaluate

$$x^{2}+13x+32$$