Solve 137.5=x^2/2-x+66

$$137.5= \frac{ { x }^{ 2 } }{ 2 } -x+66$$

Solve for x

$$x=-11$$

$$x=13$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$275=x^{2}-2x+132$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-2x+132=275$$

Subtract 275 from both sides.

$$x^{2}-2x+132-275=0$$

Subtract 275 from 132 to get -143.

$$x^{2}-2x-143=0$$

To solve the equation, factor x^{2}-2x-143 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-143$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -143.

$$1,-143 11,-13$$

Calculate the sum for each pair.

$$1-143=-142 11-13=-2$$

The solution is the pair that gives sum -2.

$$a=-13 b=11$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-13\right)\left(x+11\right)$$

To find equation solutions, solve x-13=0 and x+11=0.

$$x=13 x=-11$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$275=x^{2}-2x+132$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-2x+132=275$$

Subtract 275 from both sides.

$$x^{2}-2x+132-275=0$$

Subtract 275 from 132 to get -143.

$$x^{2}-2x-143=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-143. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-143\right)=-143$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -143.

$$1,-143 11,-13$$

Calculate the sum for each pair.

$$1-143=-142 11-13=-2$$

The solution is the pair that gives sum -2.

$$a=-13 b=11$$

Rewrite x^{2}-2x-143 as \left(x^{2}-13x\right)+\left(11x-143\right).

$$\left(x^{2}-13x\right)+\left(11x-143\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x-13\right)+11\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+11\right)$$

To find equation solutions, solve x-13=0 and x+11=0.

$$x=13 x=-11$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$275=x^{2}-2x+132$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-2x+132=275$$

Subtract 275 from both sides.

$$x^{2}-2x+132-275=0$$

Subtract 275 from 132 to get -143.

$$x^{2}-2x-143=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -143 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-143\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-143\right)}}{2}$$

Multiply -4 times -143.

$$x=\frac{-\left(-2\right)±\sqrt{4+572}}{2}$$

Add 4 to 572.

$$x=\frac{-\left(-2\right)±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-\left(-2\right)±24}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±24}{2}$$

Now solve the equation x=\frac{2±24}{2} when ± is plus. Add 2 to 24.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{2±24}{2} when ± is minus. Subtract 24 from 2.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=13 x=-11$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$275=x^{2}-2x+132$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-2x+132=275$$

Subtract 132 from both sides.

$$x^{2}-2x=275-132$$

Subtract 132 from 275 to get 143.

$$x^{2}-2x=143$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=143+1$$

Add 143 to 1.

$$x^{2}-2x+1=144$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{144}$$

Simplify.

$$x-1=12 x-1=-12$$

Add 1 to both sides of the equation.

$$x=13 x=-11$$