Solve 2a^2=-3a

$$2 a ^ { 2 } = - 3 a$$

Solve for a

$$a = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$a=0$$

Steps Using Factoring

Add 3a to both sides.

$$2a^{2}+3a=0$$

Factor out a.

$$a\left(2a+3\right)=0$$

To find equation solutions, solve a=0 and 2a+3=0.

$$a=0 a=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Add 3a to both sides.

$$2a^{2}+3a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-3±\sqrt{3^{2}}}{2\times 2}$$

Take the square root of 3^{2}.

$$a=\frac{-3±3}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{-3±3}{4}$$

Now solve the equation a=\frac{-3±3}{4} when ± is plus. Add -3 to 3.

$$a=\frac{0}{4}$$

Divide 0 by 4.

$$a=0$$

Now solve the equation a=\frac{-3±3}{4} when ± is minus. Subtract 3 from -3.

$$a=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$a=-\frac{3}{2}$$

The equation is now solved.

$$a=0 a=-\frac{3}{2}$$

Steps for Completing the Square

Add 3a to both sides.

$$2a^{2}+3a=0$$

Divide both sides by 2.

$$\frac{2a^{2}+3a}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\frac{3}{2}a=\frac{0}{2}$$

Divide 0 by 2.

$$a^{2}+\frac{3}{2}a=0$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{3}{2}a+\left(\frac{3}{4}\right)^{2}=\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{3}{2}a+\frac{9}{16}=\frac{9}{16}$$

Factor a^{2}+\frac{3}{2}a+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{3}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{3}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$a+\frac{3}{4}=\frac{3}{4} a+\frac{3}{4}=-\frac{3}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$a=0 a=-\frac{3}{2}$$