Solve 0=32+6a+2a^2

$$0 = 32 + 6 a + 2 a ^ { 2 }$$

Solve for a

$$a=\frac{-3+\sqrt{55}i}{2}\approx -1.5+3.708099244i$$

$$a=\frac{-\sqrt{55}i-3}{2}\approx -1.5-3.708099244i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$32+6a+2a^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2a^{2}+6a+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 6 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-6±\sqrt{6^{2}-4\times 2\times 32}}{2\times 2}$$

Square 6.

$$a=\frac{-6±\sqrt{36-4\times 2\times 32}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-6±\sqrt{36-8\times 32}}{2\times 2}$$

Multiply -8 times 32.

$$a=\frac{-6±\sqrt{36-256}}{2\times 2}$$

Add 36 to -256.

$$a=\frac{-6±\sqrt{-220}}{2\times 2}$$

Take the square root of -220.

$$a=\frac{-6±2\sqrt{55}i}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{-6±2\sqrt{55}i}{4}$$

Now solve the equation a=\frac{-6±2\sqrt{55}i}{4} when ± is plus. Add -6 to 2i\sqrt{55}.

$$a=\frac{-6+2\sqrt{55}i}{4}$$

Divide -6+2i\sqrt{55} by 4.

$$a=\frac{-3+\sqrt{55}i}{2}$$

Now solve the equation a=\frac{-6±2\sqrt{55}i}{4} when ± is minus. Subtract 2i\sqrt{55} from -6.

$$a=\frac{-2\sqrt{55}i-6}{4}$$

Divide -6-2i\sqrt{55} by 4.

$$a=\frac{-\sqrt{55}i-3}{2}$$

The equation is now solved.

$$a=\frac{-3+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i-3}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$32+6a+2a^{2}=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$6a+2a^{2}=-32$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2a^{2}+6a=-32$$

Divide both sides by 2.

$$\frac{2a^{2}+6a}{2}=-\frac{32}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\frac{6}{2}a=-\frac{32}{2}$$

Divide 6 by 2.

$$a^{2}+3a=-\frac{32}{2}$$

Divide -32 by 2.

$$a^{2}+3a=-16$$

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

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

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+3a+\frac{9}{4}=-16+\frac{9}{4}$$

Add -16 to \frac{9}{4}.

$$a^{2}+3a+\frac{9}{4}=-\frac{55}{4}$$

Factor a^{2}+3a+\frac{9}{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}.

$$\left(a+\frac{3}{2}\right)^{2}=-\frac{55}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{55}{4}}$$

Simplify.

$$a+\frac{3}{2}=\frac{\sqrt{55}i}{2} a+\frac{3}{2}=-\frac{\sqrt{55}i}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$a=\frac{-3+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i-3}{2}$$