Solve -2x-56=56-2x^2

$$- 2 x - 56 = 56 - 2 x ^ { 2 }$$

Solve for x

$$x=-7$$

$$x=8$$

Steps Using Factoring By Grouping

Subtract 56 from both sides.

$$-2x-56-56=-2x^{2}$$

Subtract 56 from -56 to get -112.

$$-2x-112=-2x^{2}$$

Add 2x^{2} to both sides.

$$-2x-112+2x^{2}=0$$

Divide both sides by 2.

$$-x-56+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-x-56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-56. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-56\right)=-56$$

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 -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -1.

$$a=-8 b=7$$

Rewrite x^{2}-x-56 as \left(x^{2}-8x\right)+\left(7x-56\right).

$$\left(x^{2}-8x\right)+\left(7x-56\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x-8\right)+7\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+7\right)$$

To find equation solutions, solve x-8=0 and x+7=0.

$$x=8 x=-7$$

Steps Using the Quadratic Formula

Subtract 56 from both sides.

$$-2x-56-56=-2x^{2}$$

Subtract 56 from -56 to get -112.

$$-2x-112=-2x^{2}$$

Add 2x^{2} to both sides.

$$-2x-112+2x^{2}=0$$

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.

$$2x^{2}-2x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-112\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-112\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-112\right)}}{2\times 2}$$

Multiply -8 times -112.

$$x=\frac{-\left(-2\right)±\sqrt{4+896}}{2\times 2}$$

Add 4 to 896.

$$x=\frac{-\left(-2\right)±\sqrt{900}}{2\times 2}$$

Take the square root of 900.

$$x=\frac{-\left(-2\right)±30}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±30}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±30}{4}$$

Now solve the equation x=\frac{2±30}{4} when ± is plus. Add 2 to 30.

$$x=\frac{32}{4}$$

Divide 32 by 4.

$$x=8$$

Now solve the equation x=\frac{2±30}{4} when ± is minus. Subtract 30 from 2.

$$x=-\frac{28}{4}$$

Divide -28 by 4.

$$x=-7$$

The equation is now solved.

$$x=8 x=-7$$

Steps for Completing the Square

Add 2x^{2} to both sides.

$$-2x-56+2x^{2}=56$$

Add 56 to both sides.

$$-2x+2x^{2}=56+56$$

Add 56 and 56 to get 112.

$$-2x+2x^{2}=112$$

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.

$$2x^{2}-2x=112$$

Divide both sides by 2.

$$\frac{2x^{2}-2x}{2}=\frac{112}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2}{2}\right)x=\frac{112}{2}$$

Divide -2 by 2.

$$x^{2}-x=\frac{112}{2}$$

Divide 112 by 2.

$$x^{2}-x=56$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=56+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=56+\frac{1}{4}$$

Add 56 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{225}{4}$$

Factor x^{2}-x+\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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{15}{2} x-\frac{1}{2}=-\frac{15}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=8 x=-7$$