Solve a^2+30=-8a

$$a ^ { 2 } + 30 = - 8 a$$

Solve for a

$$a=-4+\sqrt{14}i\approx -4+3.741657387i$$

$$a=-\sqrt{14}i-4\approx -4-3.741657387i$$

Steps Using the Quadratic Formula

Add 8a to both sides.

$$a^{2}+30+8a=0$$

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.

$$a^{2}+8a+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8±\sqrt{8^{2}-4\times 30}}{2}$$

Square 8.

$$a=\frac{-8±\sqrt{64-4\times 30}}{2}$$

Multiply -4 times 30.

$$a=\frac{-8±\sqrt{64-120}}{2}$$

Add 64 to -120.

$$a=\frac{-8±\sqrt{-56}}{2}$$

Take the square root of -56.

$$a=\frac{-8±2\sqrt{14}i}{2}$$

Now solve the equation a=\frac{-8±2\sqrt{14}i}{2} when ± is plus. Add -8 to 2i\sqrt{14}.

$$a=\frac{-8+2\sqrt{14}i}{2}$$

Divide -8+2i\sqrt{14} by 2.

$$a=-4+\sqrt{14}i$$

Now solve the equation a=\frac{-8±2\sqrt{14}i}{2} when ± is minus. Subtract 2i\sqrt{14} from -8.

$$a=\frac{-2\sqrt{14}i-8}{2}$$

Divide -8-2i\sqrt{14} by 2.

$$a=-\sqrt{14}i-4$$

The equation is now solved.

$$a=-4+\sqrt{14}i a=-\sqrt{14}i-4$$

Steps for Completing the Square

Add 8a to both sides.

$$a^{2}+30+8a=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$a^{2}+8a=-30$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+8a+4^{2}=-30+4^{2}$$

Square 4.

$$a^{2}+8a+16=-30+16$$

Add -30 to 16.

$$a^{2}+8a+16=-14$$

Factor a^{2}+8a+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+4\right)^{2}=-14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+4\right)^{2}}=\sqrt{-14}$$

Simplify.

$$a+4=\sqrt{14}i a+4=-\sqrt{14}i$$

Subtract 4 from both sides of the equation.

$$a=-4+\sqrt{14}i a=-\sqrt{14}i-4$$