Solve 1+9x^2=-18x+1

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

Solve for x

$$x=-2$$

$$x=0$$

Steps Using Factoring

Add 18x to both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Factor out x.

$$x\left(9x+18\right)=0$$

To find equation solutions, solve x=0 and 9x+18=0.

$$x=0 x=-2$$

Steps Using the Quadratic Formula

Add 18x to both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 18 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 18^{2}.

$$x=\frac{-18±18}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-18±18}{18}$$

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

$$x=\frac{0}{18}$$

Divide 0 by 18.

$$x=0$$

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

$$x=-\frac{36}{18}$$

Divide -36 by 18.

$$x=-2$$

The equation is now solved.

$$x=0 x=-2$$

Steps for Completing the Square

Add 18x to both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Divide both sides by 9.

$$\frac{9x^{2}+18x}{9}=\frac{0}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{18}{9}x=\frac{0}{9}$$

Divide 18 by 9.

$$x^{2}+2x=\frac{0}{9}$$

Divide 0 by 9.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=1^{2}$$

Square 1.

$$x^{2}+2x+1=1$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+1=1 x+1=-1$$

Subtract 1 from both sides of the equation.

$$x=0 x=-2$$