Solve a+2-4/2-a=59

$$a + 2 - \frac { 4 } { 2 - a } = 59$$

Solve for a

$$a = \frac{\sqrt{3009} + 59}{2} \approx 56.927176304$$

$$a = \frac{59 - \sqrt{3009}}{2} \approx 2.072823696$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -a+2.

$$\left(-a+2\right)a+\left(-a+2\right)\times 2-4=59\left(-a+2\right)$$

Use the distributive property to multiply -a+2 by a.

$$-a^{2}+2a+\left(-a+2\right)\times 2-4=59\left(-a+2\right)$$

Use the distributive property to multiply -a+2 by 2.

$$-a^{2}+2a-2a+4-4=59\left(-a+2\right)$$

Combine 2a and -2a to get 0.

$$-a^{2}+4-4=59\left(-a+2\right)$$

Subtract 4 from 4 to get 0.

$$-a^{2}=59\left(-a+2\right)$$

Use the distributive property to multiply 59 by -a+2.

$$-a^{2}=-59a+118$$

Add 59a to both sides.

$$-a^{2}+59a=118$$

Subtract 118 from both sides.

$$-a^{2}+59a-118=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 59 for b, and -118 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-59±\sqrt{59^{2}-4\left(-1\right)\left(-118\right)}}{2\left(-1\right)}$$

Square 59.

$$a=\frac{-59±\sqrt{3481-4\left(-1\right)\left(-118\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-59±\sqrt{3481+4\left(-118\right)}}{2\left(-1\right)}$$

Multiply 4 times -118.

$$a=\frac{-59±\sqrt{3481-472}}{2\left(-1\right)}$$

Add 3481 to -472.

$$a=\frac{-59±\sqrt{3009}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-59±\sqrt{3009}}{-2}$$

Now solve the equation a=\frac{-59±\sqrt{3009}}{-2} when ± is plus. Add -59 to \sqrt{3009}.

$$a=\frac{\sqrt{3009}-59}{-2}$$

Divide -59+\sqrt{3009} by -2.

$$a=\frac{59-\sqrt{3009}}{2}$$

Now solve the equation a=\frac{-59±\sqrt{3009}}{-2} when ± is minus. Subtract \sqrt{3009} from -59.

$$a=\frac{-\sqrt{3009}-59}{-2}$$

Divide -59-\sqrt{3009} by -2.

$$a=\frac{\sqrt{3009}+59}{2}$$

The equation is now solved.

$$a=\frac{59-\sqrt{3009}}{2} a=\frac{\sqrt{3009}+59}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -a+2.

$$\left(-a+2\right)a+\left(-a+2\right)\times 2-4=59\left(-a+2\right)$$

Use the distributive property to multiply -a+2 by a.

$$-a^{2}+2a+\left(-a+2\right)\times 2-4=59\left(-a+2\right)$$

Use the distributive property to multiply -a+2 by 2.

$$-a^{2}+2a-2a+4-4=59\left(-a+2\right)$$

Combine 2a and -2a to get 0.

$$-a^{2}+4-4=59\left(-a+2\right)$$

Subtract 4 from 4 to get 0.

$$-a^{2}=59\left(-a+2\right)$$

Use the distributive property to multiply 59 by -a+2.

$$-a^{2}=-59a+118$$

Add 59a to both sides.

$$-a^{2}+59a=118$$

Divide both sides by -1.

$$\frac{-a^{2}+59a}{-1}=\frac{118}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\frac{59}{-1}a=\frac{118}{-1}$$

Divide 59 by -1.

$$a^{2}-59a=\frac{118}{-1}$$

Divide 118 by -1.

$$a^{2}-59a=-118$$

Divide -59, the coefficient of the x term, by 2 to get -\frac{59}{2}. Then add the square of -\frac{59}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-59a+\left(-\frac{59}{2}\right)^{2}=-118+\left(-\frac{59}{2}\right)^{2}$$

Square -\frac{59}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-59a+\frac{3481}{4}=-118+\frac{3481}{4}$$

Add -118 to \frac{3481}{4}.

$$a^{2}-59a+\frac{3481}{4}=\frac{3009}{4}$$

Factor a^{2}-59a+\frac{3481}{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}.

$$\left(a-\frac{59}{2}\right)^{2}=\frac{3009}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{59}{2}\right)^{2}}=\sqrt{\frac{3009}{4}}$$

Simplify.

$$a-\frac{59}{2}=\frac{\sqrt{3009}}{2} a-\frac{59}{2}=-\frac{\sqrt{3009}}{2}$$

Add \frac{59}{2} to both sides of the equation.

$$a=\frac{\sqrt{3009}+59}{2} a=\frac{59-\sqrt{3009}}{2}$$