Solve 6a^2+96=6a+2

$$6 { a }^{ 2 } +96 = 6a+2$$

Solve for a

$$a=\frac{\sqrt{555}i}{6}+\frac{1}{2}\approx 0.5+3.92640633i$$

$$a=-\frac{\sqrt{555}i}{6}+\frac{1}{2}\approx 0.5-3.92640633i$$

Steps Using the Quadratic Formula

Subtract 6a from both sides.

$$6a^{2}+96-6a=2$$

Subtract 2 from both sides.

$$6a^{2}+96-6a-2=0$$

Subtract 2 from 96 to get 94.

$$6a^{2}+94-6a=0$$

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.

$$6a^{2}-6a+94=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -6 for b, and 94 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 6\times 94}}{2\times 6}$$

Square -6.

$$a=\frac{-\left(-6\right)±\sqrt{36-4\times 6\times 94}}{2\times 6}$$

Multiply -4 times 6.

$$a=\frac{-\left(-6\right)±\sqrt{36-24\times 94}}{2\times 6}$$

Multiply -24 times 94.

$$a=\frac{-\left(-6\right)±\sqrt{36-2256}}{2\times 6}$$

Add 36 to -2256.

$$a=\frac{-\left(-6\right)±\sqrt{-2220}}{2\times 6}$$

Take the square root of -2220.

$$a=\frac{-\left(-6\right)±2\sqrt{555}i}{2\times 6}$$

The opposite of -6 is 6.

$$a=\frac{6±2\sqrt{555}i}{2\times 6}$$

Multiply 2 times 6.

$$a=\frac{6±2\sqrt{555}i}{12}$$

Now solve the equation a=\frac{6±2\sqrt{555}i}{12} when ± is plus. Add 6 to 2i\sqrt{555}.

$$a=\frac{6+2\sqrt{555}i}{12}$$

Divide 6+2i\sqrt{555} by 12.

$$a=\frac{\sqrt{555}i}{6}+\frac{1}{2}$$

Now solve the equation a=\frac{6±2\sqrt{555}i}{12} when ± is minus. Subtract 2i\sqrt{555} from 6.

$$a=\frac{-2\sqrt{555}i+6}{12}$$

Divide 6-2i\sqrt{555} by 12.

$$a=-\frac{\sqrt{555}i}{6}+\frac{1}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{555}i}{6}+\frac{1}{2} a=-\frac{\sqrt{555}i}{6}+\frac{1}{2}$$

Steps for Completing the Square

Subtract 6a from both sides.

$$6a^{2}+96-6a=2$$

Subtract 96 from both sides.

$$6a^{2}-6a=2-96$$

Subtract 96 from 2 to get -94.

$$6a^{2}-6a=-94$$

Divide both sides by 6.

$$\frac{6a^{2}-6a}{6}=-\frac{94}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$a^{2}+\left(-\frac{6}{6}\right)a=-\frac{94}{6}$$

Divide -6 by 6.

$$a^{2}-a=-\frac{94}{6}$$

Reduce the fraction \frac{-94}{6} to lowest terms by extracting and canceling out 2.

$$a^{2}-a=-\frac{47}{3}$$

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.

$$a^{2}-a+\left(-\frac{1}{2}\right)^{2}=-\frac{47}{3}+\left(-\frac{1}{2}\right)^{2}$$

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

$$a^{2}-a+\frac{1}{4}=-\frac{47}{3}+\frac{1}{4}$$

Add -\frac{47}{3} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-a+\frac{1}{4}=-\frac{185}{12}$$

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}.

$$\left(a-\frac{1}{2}\right)^{2}=-\frac{185}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{185}{12}}$$

Simplify.

$$a-\frac{1}{2}=\frac{\sqrt{555}i}{6} a-\frac{1}{2}=-\frac{\sqrt{555}i}{6}$$

Add \frac{1}{2} to both sides of the equation.

$$a=\frac{\sqrt{555}i}{6}+\frac{1}{2} a=-\frac{\sqrt{555}i}{6}+\frac{1}{2}$$