Solve 9x^2+18x+1

$$9 { x }^{ 2 } +18x+1$$

Factor

$$9\left(x-\left(-\frac{2\sqrt{2}}{3}-1\right)\right)\left(x-\left(\frac{2\sqrt{2}}{3}-1\right)\right)$$

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.

$$9x^{2}+18x+1=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\times 9}}{2\times 9}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 9}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-18±\sqrt{324-36}}{2\times 9}$$

Add 324 to -36.

$$x=\frac{-18±\sqrt{288}}{2\times 9}$$

Take the square root of 288.

$$x=\frac{-18±12\sqrt{2}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-18±12\sqrt{2}}{18}$$

Now solve the equation x=\frac{-18±12\sqrt{2}}{18} when ± is plus. Add -18 to 12\sqrt{2}.

$$x=\frac{12\sqrt{2}-18}{18}$$

Divide -18+12\sqrt{2} by 18.

$$x=\frac{2\sqrt{2}}{3}-1$$

Now solve the equation x=\frac{-18±12\sqrt{2}}{18} when ± is minus. Subtract 12\sqrt{2} from -18.

$$x=\frac{-12\sqrt{2}-18}{18}$$

Divide -18-12\sqrt{2} by 18.

$$x=-\frac{2\sqrt{2}}{3}-1$$

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

$$9x^{2}+18x+1=9\left(x-\left(\frac{2\sqrt{2}}{3}-1\right)\right)\left(x-\left(-\frac{2\sqrt{2}}{3}-1\right)\right)$$

Evaluate

$$9x^{2}+18x+1$$