Solve 9x(x+9)=0

$$9 x ( x + 9 ) = 0$$

Solve for x

$$x=-9$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 9x by x+9.

$$9x^{2}+81x=0$$

Factor out x.

$$x\left(9x+81\right)=0$$

To find equation solutions, solve x=0 and 9x+81=0.

$$x=0 x=-9$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9x by x+9.

$$9x^{2}+81x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 81 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-81±\sqrt{81^{2}}}{2\times 9}$$

Take the square root of 81^{2}.

$$x=\frac{-81±81}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-81±81}{18}$$

Now solve the equation x=\frac{-81±81}{18} when ± is plus. Add -81 to 81.

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

Divide 0 by 18.

$$x=0$$

Now solve the equation x=\frac{-81±81}{18} when ± is minus. Subtract 81 from -81.

$$x=-\frac{162}{18}$$

Divide -162 by 18.

$$x=-9$$

The equation is now solved.

$$x=0 x=-9$$

Steps for Completing the Square

Use the distributive property to multiply 9x by x+9.

$$9x^{2}+81x=0$$

Divide both sides by 9.

$$\frac{9x^{2}+81x}{9}=\frac{0}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{81}{9}x=\frac{0}{9}$$

Divide 81 by 9.

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

Divide 0 by 9.

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

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=\frac{81}{4}$$

Factor x^{2}+9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{9}{2} x+\frac{9}{2}=-\frac{9}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=0 x=-9$$