Solve 2a+2-4/a=8/a

$$2 a + 2 - \frac { 4 } { a } = \frac { 8 } { a }$$

Solve for a

$$a=-3$$

$$a=2$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$2aa+a\times 2-4=8$$

Multiply a and a to get a^{2}.

$$2a^{2}+a\times 2-4=8$$

Subtract 8 from both sides.

$$2a^{2}+a\times 2-4-8=0$$

Subtract 8 from -4 to get -12.

$$2a^{2}+a\times 2-12=0$$

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.

$$2a^{2}+2a-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2±\sqrt{2^{2}-4\times 2\left(-12\right)}}{2\times 2}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\times 2\left(-12\right)}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-2±\sqrt{4-8\left(-12\right)}}{2\times 2}$$

Multiply -8 times -12.

$$a=\frac{-2±\sqrt{4+96}}{2\times 2}$$

Add 4 to 96.

$$a=\frac{-2±\sqrt{100}}{2\times 2}$$

Take the square root of 100.

$$a=\frac{-2±10}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{-2±10}{4}$$

Now solve the equation a=\frac{-2±10}{4} when ± is plus. Add -2 to 10.

$$a=\frac{8}{4}$$

Divide 8 by 4.

$$a=2$$

Now solve the equation a=\frac{-2±10}{4} when ± is minus. Subtract 10 from -2.

$$a=-\frac{12}{4}$$

Divide -12 by 4.

$$a=-3$$

The equation is now solved.

$$a=2 a=-3$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$2aa+a\times 2-4=8$$

Multiply a and a to get a^{2}.

$$2a^{2}+a\times 2-4=8$$

Add 4 to both sides.

$$2a^{2}+a\times 2=8+4$$

Add 8 and 4 to get 12.

$$2a^{2}+a\times 2=12$$

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.

$$2a^{2}+2a=12$$

Divide both sides by 2.

$$\frac{2a^{2}+2a}{2}=\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\frac{2}{2}a=\frac{12}{2}$$

Divide 2 by 2.

$$a^{2}+a=\frac{12}{2}$$

Divide 12 by 2.

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

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}.

$$a^{2}+a+\frac{1}{4}=\frac{25}{4}$$

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}.

$$\left(a+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$a+\frac{1}{2}=\frac{5}{2} a+\frac{1}{2}=-\frac{5}{2}$$

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

$$a=2 a=-3$$