Solve x^2+12x+14

$${ x }^{ 2 } +12x+14$$

Factor

$$\left(x-\left(-\sqrt{22}-6\right)\right)\left(x-\left(\sqrt{22}-6\right)\right)$$

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.

$$x^{2}+12x+14=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 14}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 14}}{2}$$

Multiply -4 times 14.

$$x=\frac{-12±\sqrt{144-56}}{2}$$

Add 144 to -56.

$$x=\frac{-12±\sqrt{88}}{2}$$

Take the square root of 88.

$$x=\frac{-12±2\sqrt{22}}{2}$$

Now solve the equation x=\frac{-12±2\sqrt{22}}{2} when ± is plus. Add -12 to 2\sqrt{22}.

$$x=\frac{2\sqrt{22}-12}{2}$$

Divide -12+2\sqrt{22} by 2.

$$x=\sqrt{22}-6$$

Now solve the equation x=\frac{-12±2\sqrt{22}}{2} when ± is minus. Subtract 2\sqrt{22} from -12.

$$x=\frac{-2\sqrt{22}-12}{2}$$

Divide -12-2\sqrt{22} by 2.

$$x=-\sqrt{22}-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6+\sqrt{22} for x_{1} and -6-\sqrt{22} for x_{2}.

$$x^{2}+12x+14=\left(x-\left(\sqrt{22}-6\right)\right)\left(x-\left(-\sqrt{22}-6\right)\right)$$

Evaluate

$$x^{2}+12x+14$$