Solve [-(a+2)^2-a]*3=8

$$[ - ( a + 2 ) ^ { 2 } - a ] \times 3 = 8$$

Solve for a

$$a=\frac{\sqrt{15}i}{6}-\frac{5}{2}\approx -2.5+0.645497224i$$

$$a=-\frac{\sqrt{15}i}{6}-\frac{5}{2}\approx -2.5-0.645497224i$$

Steps Using the Quadratic Formula

Divide both sides by 3.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+2\right)^{2}.

$$-\left(a^{2}+4a+4\right)-a=\frac{8}{3}$$

To find the opposite of a^{2}+4a+4, find the opposite of each term.

$$-a^{2}-4a-4-a=\frac{8}{3}$$

Combine -4a and -a to get -5a.

$$-a^{2}-5a-4=\frac{8}{3}$$

Subtract \frac{8}{3} from both sides.

$$-a^{2}-5a-4-\frac{8}{3}=0$$

Subtract \frac{8}{3} from -4 to get -\frac{20}{3}.

$$-a^{2}-5a-\frac{20}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -5 for b, and -\frac{20}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\left(-\frac{20}{3}\right)}}{2\left(-1\right)}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\left(-\frac{20}{3}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-5\right)±\sqrt{25+4\left(-\frac{20}{3}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{20}{3}.

$$a=\frac{-\left(-5\right)±\sqrt{25-\frac{80}{3}}}{2\left(-1\right)}$$

Add 25 to -\frac{80}{3}.

$$a=\frac{-\left(-5\right)±\sqrt{-\frac{5}{3}}}{2\left(-1\right)}$$

Take the square root of -\frac{5}{3}.

$$a=\frac{-\left(-5\right)±\frac{\sqrt{15}i}{3}}{2\left(-1\right)}$$

The opposite of -5 is 5.

$$a=\frac{5±\frac{\sqrt{15}i}{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{5±\frac{\sqrt{15}i}{3}}{-2}$$

Now solve the equation a=\frac{5±\frac{\sqrt{15}i}{3}}{-2} when ± is plus. Add 5 to \frac{i\sqrt{15}}{3}.

$$a=\frac{\frac{\sqrt{15}i}{3}+5}{-2}$$

Divide 5+\frac{i\sqrt{15}}{3} by -2.

$$a=-\frac{\sqrt{15}i}{6}-\frac{5}{2}$$

Now solve the equation a=\frac{5±\frac{\sqrt{15}i}{3}}{-2} when ± is minus. Subtract \frac{i\sqrt{15}}{3} from 5.

$$a=\frac{-\frac{\sqrt{15}i}{3}+5}{-2}$$

Divide 5-\frac{i\sqrt{15}}{3} by -2.

$$a=\frac{\sqrt{15}i}{6}-\frac{5}{2}$$

The equation is now solved.

$$a=-\frac{\sqrt{15}i}{6}-\frac{5}{2} a=\frac{\sqrt{15}i}{6}-\frac{5}{2}$$

Steps for Completing the Square

Divide both sides by 3.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+2\right)^{2}.

$$-\left(a^{2}+4a+4\right)-a=\frac{8}{3}$$

To find the opposite of a^{2}+4a+4, find the opposite of each term.

$$-a^{2}-4a-4-a=\frac{8}{3}$$

Combine -4a and -a to get -5a.

$$-a^{2}-5a-4=\frac{8}{3}$$

Add 4 to both sides.

$$-a^{2}-5a=\frac{8}{3}+4$$

Add \frac{8}{3} and 4 to get \frac{20}{3}.

$$-a^{2}-5a=\frac{20}{3}$$

Divide both sides by -1.

$$\frac{-a^{2}-5a}{-1}=\frac{\frac{20}{3}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\left(-\frac{5}{-1}\right)a=\frac{\frac{20}{3}}{-1}$$

Divide -5 by -1.

$$a^{2}+5a=\frac{\frac{20}{3}}{-1}$$

Divide \frac{20}{3} by -1.

$$a^{2}+5a=-\frac{20}{3}$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+5a+\left(\frac{5}{2}\right)^{2}=-\frac{20}{3}+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+5a+\frac{25}{4}=-\frac{20}{3}+\frac{25}{4}$$

Add -\frac{20}{3} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+5a+\frac{25}{4}=-\frac{5}{12}$$

Factor a^{2}+5a+\frac{25}{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}.

$$\left(a+\frac{5}{2}\right)^{2}=-\frac{5}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{5}{12}}$$

Simplify.

$$a+\frac{5}{2}=\frac{\sqrt{15}i}{6} a+\frac{5}{2}=-\frac{\sqrt{15}i}{6}$$

Subtract \frac{5}{2} from both sides of the equation.

$$a=\frac{\sqrt{15}i}{6}-\frac{5}{2} a=-\frac{\sqrt{15}i}{6}-\frac{5}{2}$$