Solve -4x^2=9x

$$- 4 x ^ { 2 } = 9 x$$

Solve for x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=0$$

Steps Using Factoring

Subtract 9x from both sides.

$$-4x^{2}-9x=0$$

Factor out x.

$$x\left(-4x-9\right)=0$$

To find equation solutions, solve x=0 and -4x-9=0.

$$x=0 x=-\frac{9}{4}$$

Steps Using the Quadratic Formula

Subtract 9x from both sides.

$$-4x^{2}-9x=0$$

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

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

Take the square root of \left(-9\right)^{2}.

$$x=\frac{-\left(-9\right)±9}{2\left(-4\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±9}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{9±9}{-8}$$

Now solve the equation x=\frac{9±9}{-8} when ± is plus. Add 9 to 9.

$$x=\frac{18}{-8}$$

Reduce the fraction \frac{18}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{4}$$

Now solve the equation x=\frac{9±9}{-8} when ± is minus. Subtract 9 from 9.

$$x=\frac{0}{-8}$$

Divide 0 by -8.

$$x=0$$

The equation is now solved.

$$x=-\frac{9}{4} x=0$$

Steps for Completing the Square

Subtract 9x from both sides.

$$-4x^{2}-9x=0$$

Divide both sides by -4.

$$\frac{-4x^{2}-9x}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{9}{-4}\right)x=\frac{0}{-4}$$

Divide -9 by -4.

$$x^{2}+\frac{9}{4}x=\frac{0}{-4}$$

Divide 0 by -4.

$$x^{2}+\frac{9}{4}x=0$$

Divide \frac{9}{4}, the coefficient of the x term, by 2 to get \frac{9}{8}. Then add the square of \frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{4}x+\left(\frac{9}{8}\right)^{2}=\left(\frac{9}{8}\right)^{2}$$

Square \frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{81}{64}$$

Factor x^{2}+\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{8}\right)^{2}=\frac{81}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Simplify.

$$x+\frac{9}{8}=\frac{9}{8} x+\frac{9}{8}=-\frac{9}{8}$$

Subtract \frac{9}{8} from both sides of the equation.

$$x=0 x=-\frac{9}{4}$$