Solve 3(-a^2-2a+3)/2+3a/2=8

$$\frac { 3 ( - a ^ { 2 } - 2 a + 3 ) } { 2 } + \frac { 3 a } { 2 } = 8$$

Solve for a

$$a=\frac{5\sqrt{3}i}{6}-\frac{1}{2}\approx -0.5+1.443375673i$$

$$a=-\frac{5\sqrt{3}i}{6}-\frac{1}{2}\approx -0.5-1.443375673i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$3\left(-a^{2}-2a+3\right)+3a=16$$

Use the distributive property to multiply 3 by -a^{2}-2a+3.

$$3\left(-a^{2}\right)-6a+9+3a=16$$

Combine -6a and 3a to get -3a.

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

Subtract 16 from both sides.

$$3\left(-a^{2}\right)-3a+9-16=0$$

Subtract 16 from 9 to get -7.

$$3\left(-a^{2}\right)-3a-7=0$$

Multiply 3 and -1 to get -3.

$$-3a^{2}-3a-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -3 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-3\right)\left(-7\right)}}{2\left(-3\right)}$$

Square -3.

$$a=\frac{-\left(-3\right)±\sqrt{9-4\left(-3\right)\left(-7\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$a=\frac{-\left(-3\right)±\sqrt{9+12\left(-7\right)}}{2\left(-3\right)}$$

Multiply 12 times -7.

$$a=\frac{-\left(-3\right)±\sqrt{9-84}}{2\left(-3\right)}$$

Add 9 to -84.

$$a=\frac{-\left(-3\right)±\sqrt{-75}}{2\left(-3\right)}$$

Take the square root of -75.

$$a=\frac{-\left(-3\right)±5\sqrt{3}i}{2\left(-3\right)}$$

The opposite of -3 is 3.

$$a=\frac{3±5\sqrt{3}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{3±5\sqrt{3}i}{-6}$$

Now solve the equation a=\frac{3±5\sqrt{3}i}{-6} when ± is plus. Add 3 to 5i\sqrt{3}.

$$a=\frac{3+5\sqrt{3}i}{-6}$$

Divide 3+5i\sqrt{3} by -6.

$$a=-\frac{5\sqrt{3}i}{6}-\frac{1}{2}$$

Now solve the equation a=\frac{3±5\sqrt{3}i}{-6} when ± is minus. Subtract 5i\sqrt{3} from 3.

$$a=\frac{-5\sqrt{3}i+3}{-6}$$

Divide 3-5i\sqrt{3} by -6.

$$a=\frac{5\sqrt{3}i}{6}-\frac{1}{2}$$

The equation is now solved.

$$a=-\frac{5\sqrt{3}i}{6}-\frac{1}{2} a=\frac{5\sqrt{3}i}{6}-\frac{1}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$3\left(-a^{2}-2a+3\right)+3a=16$$

Use the distributive property to multiply 3 by -a^{2}-2a+3.

$$3\left(-a^{2}\right)-6a+9+3a=16$$

Combine -6a and 3a to get -3a.

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

Subtract 9 from both sides.

$$3\left(-a^{2}\right)-3a=16-9$$

Subtract 9 from 16 to get 7.

$$3\left(-a^{2}\right)-3a=7$$

Multiply 3 and -1 to get -3.

$$-3a^{2}-3a=7$$

Divide both sides by -3.

$$\frac{-3a^{2}-3a}{-3}=\frac{7}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$a^{2}+\left(-\frac{3}{-3}\right)a=\frac{7}{-3}$$

Divide -3 by -3.

$$a^{2}+a=\frac{7}{-3}$$

Divide 7 by -3.

$$a^{2}+a=-\frac{7}{3}$$

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

$$a^{2}+a+\left(\frac{1}{2}\right)^{2}=-\frac{7}{3}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+a+\frac{1}{4}=-\frac{7}{3}+\frac{1}{4}$$

Add -\frac{7}{3} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+a+\frac{1}{4}=-\frac{25}{12}$$

Factor a^{2}+a+\frac{1}{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}.

$$\left(a+\frac{1}{2}\right)^{2}=-\frac{25}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{25}{12}}$$

Simplify.

$$a+\frac{1}{2}=\frac{5\sqrt{3}i}{6} a+\frac{1}{2}=-\frac{5\sqrt{3}i}{6}$$

Subtract \frac{1}{2} from both sides of the equation.

$$a=\frac{5\sqrt{3}i}{6}-\frac{1}{2} a=-\frac{5\sqrt{3}i}{6}-\frac{1}{2}$$