Solve -5/2+1/2y^2-2y-5/2=0

$$- \frac { 5 } { 2 } + \frac { 1 } { 2 } y ^ { 2 } - 2 y - \frac { 5 } { 2 } = 0$$

Solve for y

$$y=\sqrt{14}+2\approx 5.741657387$$

$$y=2-\sqrt{14}\approx -1.741657387$$

Steps Using the Quadratic Formula

Subtract \frac{5}{2} from -\frac{5}{2} to get -5.

$$-5+\frac{1}{2}y^{2}-2y=0$$

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.

$$\frac{1}{2}y^{2}-2y-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -2 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times \frac{1}{2}\left(-5\right)}}{2\times \frac{1}{2}}$$

Square -2.

$$y=\frac{-\left(-2\right)±\sqrt{4-4\times \frac{1}{2}\left(-5\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$y=\frac{-\left(-2\right)±\sqrt{4-2\left(-5\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -5.

$$y=\frac{-\left(-2\right)±\sqrt{4+10}}{2\times \frac{1}{2}}$$

Add 4 to 10.

$$y=\frac{-\left(-2\right)±\sqrt{14}}{2\times \frac{1}{2}}$$

The opposite of -2 is 2.

$$y=\frac{2±\sqrt{14}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$y=\frac{2±\sqrt{14}}{1}$$

Now solve the equation y=\frac{2±\sqrt{14}}{1} when ± is plus. Add 2 to \sqrt{14}.

$$y=\frac{\sqrt{14}+2}{1}$$

Divide 2+\sqrt{14} by 1.

$$y=\sqrt{14}+2$$

Now solve the equation y=\frac{2±\sqrt{14}}{1} when ± is minus. Subtract \sqrt{14} from 2.

$$y=\frac{2-\sqrt{14}}{1}$$

Divide 2-\sqrt{14} by 1.

$$y=2-\sqrt{14}$$

The equation is now solved.

$$y=\sqrt{14}+2 y=2-\sqrt{14}$$

Steps for Completing the Square

Subtract \frac{5}{2} from -\frac{5}{2} to get -5.

$$-5+\frac{1}{2}y^{2}-2y=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$\frac{1}{2}y^{2}-2y=5$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}y^{2}-2y}{\frac{1}{2}}=\frac{5}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$y^{2}+\left(-\frac{2}{\frac{1}{2}}\right)y=\frac{5}{\frac{1}{2}}$$

Divide -2 by \frac{1}{2} by multiplying -2 by the reciprocal of \frac{1}{2}.

$$y^{2}-4y=\frac{5}{\frac{1}{2}}$$

Divide 5 by \frac{1}{2} by multiplying 5 by the reciprocal of \frac{1}{2}.

$$y^{2}-4y=10$$

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

$$y^{2}-4y+\left(-2\right)^{2}=10+\left(-2\right)^{2}$$

Square -2.

$$y^{2}-4y+4=10+4$$

Add 10 to 4.

$$y^{2}-4y+4=14$$

Factor y^{2}-4y+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}.

$$\left(y-2\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-2\right)^{2}}=\sqrt{14}$$

Simplify.

$$y-2=\sqrt{14} y-2=-\sqrt{14}$$

Add 2 to both sides of the equation.

$$y=\sqrt{14}+2 y=2-\sqrt{14}$$