Solve -a^2/a^2-9+3/a+3=3a/3a-9

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

Solve for a

$$a=-\frac{3\sqrt{2}i}{2}\approx -0-2.121320344i$$

$$a=\frac{3\sqrt{2}i}{2}\approx 2.121320344i$$

Steps by Finding Square Root

Variable a cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 3\left(a-3\right)\left(a+3\right), the least common multiple of a^{2}-9,a+3,3a-9.

$$-3a^{2}+\left(3a-9\right)\times 3=\left(a+3\right)\times 3a$$

Use the distributive property to multiply 3a-9 by 3.

$$-3a^{2}+9a-27=\left(a+3\right)\times 3a$$

Use the distributive property to multiply a+3 by 3.

$$-3a^{2}+9a-27=\left(3a+9\right)a$$

Use the distributive property to multiply 3a+9 by a.

$$-3a^{2}+9a-27=3a^{2}+9a$$

Subtract 3a^{2} from both sides.

$$-3a^{2}+9a-27-3a^{2}=9a$$

Combine -3a^{2} and -3a^{2} to get -6a^{2}.

$$-6a^{2}+9a-27=9a$$

Subtract 9a from both sides.

$$-6a^{2}+9a-27-9a=0$$

Combine 9a and -9a to get 0.

$$-6a^{2}-27=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$-6a^{2}=27$$

Divide both sides by -6.

$$a^{2}=\frac{27}{-6}$$

Reduce the fraction \frac{27}{-6} to lowest terms by extracting and canceling out 3.

$$a^{2}=-\frac{9}{2}$$

The equation is now solved.

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

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 3\left(a-3\right)\left(a+3\right), the least common multiple of a^{2}-9,a+3,3a-9.

$$-3a^{2}+\left(3a-9\right)\times 3=\left(a+3\right)\times 3a$$

Use the distributive property to multiply 3a-9 by 3.

$$-3a^{2}+9a-27=\left(a+3\right)\times 3a$$

Use the distributive property to multiply a+3 by 3.

$$-3a^{2}+9a-27=\left(3a+9\right)a$$

Use the distributive property to multiply 3a+9 by a.

$$-3a^{2}+9a-27=3a^{2}+9a$$

Subtract 3a^{2} from both sides.

$$-3a^{2}+9a-27-3a^{2}=9a$$

Combine -3a^{2} and -3a^{2} to get -6a^{2}.

$$-6a^{2}+9a-27=9a$$

Subtract 9a from both sides.

$$-6a^{2}+9a-27-9a=0$$

Combine 9a and -9a to get 0.

$$-6a^{2}-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 0 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}-4\left(-6\right)\left(-27\right)}}{2\left(-6\right)}$$

Square 0.

$$a=\frac{0±\sqrt{-4\left(-6\right)\left(-27\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$a=\frac{0±\sqrt{24\left(-27\right)}}{2\left(-6\right)}$$

Multiply 24 times -27.

$$a=\frac{0±\sqrt{-648}}{2\left(-6\right)}$$

Take the square root of -648.

$$a=\frac{0±18\sqrt{2}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$a=\frac{0±18\sqrt{2}i}{-12}$$

Now solve the equation a=\frac{0±18\sqrt{2}i}{-12} when ± is plus.

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

Now solve the equation a=\frac{0±18\sqrt{2}i}{-12} when ± is minus.

$$a=\frac{3\sqrt{2}i}{2}$$

The equation is now solved.

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