Solve 81x^2+18x^2+x=0

$$81 x ^ { 2 } + 18 x ^ { 2 } + x = 0$$

Solve for x

$$x=-\frac{1}{99}\approx -0.01010101$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(81x+18x+1\right)=0$$

To find equation solutions, solve x=0 and 81x+18x+1=0.

$$x=0 x=-\frac{1}{99}$$

Steps Using the Quadratic Formula

Combine 81x^{2} and 18x^{2} to get 99x^{2}.

$$99x^{2}+x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 99 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}}}{2\times 99}$$

Take the square root of 1^{2}.

$$x=\frac{-1±1}{2\times 99}$$

Multiply 2 times 99.

$$x=\frac{-1±1}{198}$$

Now solve the equation x=\frac{-1±1}{198} when ± is plus. Add -1 to 1.

$$x=\frac{0}{198}$$

Divide 0 by 198.

$$x=0$$

Now solve the equation x=\frac{-1±1}{198} when ± is minus. Subtract 1 from -1.

$$x=-\frac{2}{198}$$

Reduce the fraction \frac{-2}{198} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{99}$$

The equation is now solved.

$$x=0 x=-\frac{1}{99}$$

Steps for Completing the Square

Combine 81x^{2} and 18x^{2} to get 99x^{2}.

$$99x^{2}+x=0$$

Divide both sides by 99.

$$\frac{99x^{2}+x}{99}=\frac{0}{99}$$

Dividing by 99 undoes the multiplication by 99.

$$x^{2}+\frac{1}{99}x=\frac{0}{99}$$

Divide 0 by 99.

$$x^{2}+\frac{1}{99}x=0$$

Divide \frac{1}{99}, the coefficient of the x term, by 2 to get \frac{1}{198}. Then add the square of \frac{1}{198} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{99}x+\left(\frac{1}{198}\right)^{2}=\left(\frac{1}{198}\right)^{2}$$

Square \frac{1}{198} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{99}x+\frac{1}{39204}=\frac{1}{39204}$$

Factor x^{2}+\frac{1}{99}x+\frac{1}{39204}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{198}\right)^{2}=\frac{1}{39204}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{198}\right)^{2}}=\sqrt{\frac{1}{39204}}$$

Simplify.

$$x+\frac{1}{198}=\frac{1}{198} x+\frac{1}{198}=-\frac{1}{198}$$

Subtract \frac{1}{198} from both sides of the equation.

$$x=0 x=-\frac{1}{99}$$