Solve 8x^2+12x+1

$$8 { x }^{ 2 } +12x+1$$

Factor

$$8\left(x-\frac{-\sqrt{7}-3}{4}\right)\left(x-\frac{\sqrt{7}-3}{4}\right)$$

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.

$$8x^{2}+12x+1=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 8}}{2\times 8}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 8}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-12±\sqrt{144-32}}{2\times 8}$$

Add 144 to -32.

$$x=\frac{-12±\sqrt{112}}{2\times 8}$$

Take the square root of 112.

$$x=\frac{-12±4\sqrt{7}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-12±4\sqrt{7}}{16}$$

Now solve the equation x=\frac{-12±4\sqrt{7}}{16} when ± is plus. Add -12 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}-12}{16}$$

Divide -12+4\sqrt{7} by 16.

$$x=\frac{\sqrt{7}-3}{4}$$

Now solve the equation x=\frac{-12±4\sqrt{7}}{16} when ± is minus. Subtract 4\sqrt{7} from -12.

$$x=\frac{-4\sqrt{7}-12}{16}$$

Divide -12-4\sqrt{7} by 16.

$$x=\frac{-\sqrt{7}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{7}}{4} for x_{1} and \frac{-3-\sqrt{7}}{4} for x_{2}.

$$8x^{2}+12x+1=8\left(x-\frac{\sqrt{7}-3}{4}\right)\left(x-\frac{-\sqrt{7}-3}{4}\right)$$

Evaluate

$$8x^{2}+12x+1$$