Solve x^2+20x+22

$${ x }^{ 2 } +20x+22$$

Factor

$$\left(x-\left(-\sqrt{78}-10\right)\right)\left(x-\left(\sqrt{78}-10\right)\right)$$

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.

$$x^{2}+20x+22=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\times 22}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 22}}{2}$$

Multiply -4 times 22.

$$x=\frac{-20±\sqrt{400-88}}{2}$$

Add 400 to -88.

$$x=\frac{-20±\sqrt{312}}{2}$$

Take the square root of 312.

$$x=\frac{-20±2\sqrt{78}}{2}$$

Now solve the equation x=\frac{-20±2\sqrt{78}}{2} when ± is plus. Add -20 to 2\sqrt{78}.

$$x=\frac{2\sqrt{78}-20}{2}$$

Divide -20+2\sqrt{78} by 2.

$$x=\sqrt{78}-10$$

Now solve the equation x=\frac{-20±2\sqrt{78}}{2} when ± is minus. Subtract 2\sqrt{78} from -20.

$$x=\frac{-2\sqrt{78}-20}{2}$$

Divide -20-2\sqrt{78} by 2.

$$x=-\sqrt{78}-10$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -10+\sqrt{78} for x_{1} and -10-\sqrt{78} for x_{2}.

$$x^{2}+20x+22=\left(x-\left(\sqrt{78}-10\right)\right)\left(x-\left(-\sqrt{78}-10\right)\right)$$

Evaluate

$$x^{2}+20x+22$$