Solve 0=a^2+1a+6

$$0 = a ^ { 2 } + 1 a + 6$$

Solve for a

$$a=\frac{-1+\sqrt{23}i}{2}\approx -0.5+2.397915762i$$

$$a=\frac{-\sqrt{23}i-1}{2}\approx -0.5-2.397915762i$$

Steps Using the Quadratic Formula

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

$$a^{2}+1a+6=0$$

Reorder the terms.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-1±\sqrt{1^{2}-4\times 6}}{2}$$

Square 1.

$$a=\frac{-1±\sqrt{1-4\times 6}}{2}$$

Multiply -4 times 6.

$$a=\frac{-1±\sqrt{1-24}}{2}$$

Add 1 to -24.

$$a=\frac{-1±\sqrt{-23}}{2}$$

Take the square root of -23.

$$a=\frac{-1±\sqrt{23}i}{2}$$

Now solve the equation a=\frac{-1±\sqrt{23}i}{2} when ± is plus. Add -1 to i\sqrt{23}.

$$a=\frac{-1+\sqrt{23}i}{2}$$

Now solve the equation a=\frac{-1±\sqrt{23}i}{2} when ± is minus. Subtract i\sqrt{23} from -1.

$$a=\frac{-\sqrt{23}i-1}{2}$$

The equation is now solved.

$$a=\frac{-1+\sqrt{23}i}{2} a=\frac{-\sqrt{23}i-1}{2}$$

Steps for Completing the Square

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

$$a^{2}+1a+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$a^{2}+1a=-6$$

Reorder the terms.

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

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.

$$a^{2}+a+\left(\frac{1}{2}\right)^{2}=-6+\left(\frac{1}{2}\right)^{2}$$

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

$$a^{2}+a+\frac{1}{4}=-6+\frac{1}{4}$$

Add -6 to \frac{1}{4}.

$$a^{2}+a+\frac{1}{4}=-\frac{23}{4}$$

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}.

$$\left(a+\frac{1}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$a+\frac{1}{2}=\frac{\sqrt{23}i}{2} a+\frac{1}{2}=-\frac{\sqrt{23}i}{2}$$

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

$$a=\frac{-1+\sqrt{23}i}{2} a=\frac{-\sqrt{23}i-1}{2}$$