Solve a^2+2/2a-2/4=4^2a/2

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

Solve for a

$$a = \frac{\sqrt{969} + 31}{2} \approx 31.064382416$$

$$a=\frac{31-\sqrt{969}}{2}\approx -0.064382416$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 4,2.

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

Divide 2 by 2 to get 1.

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

Calculate 4 to the power of 2 and get 16.

$$a^{2}+1a-2=2\times 16a$$

Multiply 2 and 16 to get 32.

$$a^{2}+1a-2=32a$$

Subtract 32a from both sides.

$$a^{2}+1a-2-32a=0$$

Combine 1a and -32a to get -31a.

$$a^{2}-31a-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -31 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\left(-2\right)}}{2}$$

Square -31.

$$a=\frac{-\left(-31\right)±\sqrt{961-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$a=\frac{-\left(-31\right)±\sqrt{961+8}}{2}$$

Add 961 to 8.

$$a=\frac{-\left(-31\right)±\sqrt{969}}{2}$$

The opposite of -31 is 31.

$$a=\frac{31±\sqrt{969}}{2}$$

Now solve the equation a=\frac{31±\sqrt{969}}{2} when ± is plus. Add 31 to \sqrt{969}.

$$a=\frac{\sqrt{969}+31}{2}$$

Now solve the equation a=\frac{31±\sqrt{969}}{2} when ± is minus. Subtract \sqrt{969} from 31.

$$a=\frac{31-\sqrt{969}}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{969}+31}{2} a=\frac{31-\sqrt{969}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 4,2.

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

Divide 2 by 2 to get 1.

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

Calculate 4 to the power of 2 and get 16.

$$a^{2}+1a-2=2\times 16a$$

Multiply 2 and 16 to get 32.

$$a^{2}+1a-2=32a$$

Subtract 32a from both sides.

$$a^{2}+1a-2-32a=0$$

Combine 1a and -32a to get -31a.

$$a^{2}-31a-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$a^{2}-31a=2$$

Divide -31, the coefficient of the x term, by 2 to get -\frac{31}{2}. Then add the square of -\frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-31a+\left(-\frac{31}{2}\right)^{2}=2+\left(-\frac{31}{2}\right)^{2}$$

Square -\frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-31a+\frac{961}{4}=2+\frac{961}{4}$$

Add 2 to \frac{961}{4}.

$$a^{2}-31a+\frac{961}{4}=\frac{969}{4}$$

Factor a^{2}-31a+\frac{961}{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}.

$$\left(a-\frac{31}{2}\right)^{2}=\frac{969}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{31}{2}\right)^{2}}=\sqrt{\frac{969}{4}}$$

Simplify.

$$a-\frac{31}{2}=\frac{\sqrt{969}}{2} a-\frac{31}{2}=-\frac{\sqrt{969}}{2}$$

Add \frac{31}{2} to both sides of the equation.

$$a=\frac{\sqrt{969}+31}{2} a=\frac{31-\sqrt{969}}{2}$$