Solve x^2-x-20=2x+8

$$x ^ { 2 } - x - 20 = 2 x + 8$$

Solve for x

$$x=-4$$

$$x=7$$

Steps Using Factoring

Subtract 2x from both sides.

$$x^{2}-x-20-2x=8$$

Combine -x and -2x to get -3x.

$$x^{2}-3x-20=8$$

Subtract 8 from both sides.

$$x^{2}-3x-20-8=0$$

Subtract 8 from -20 to get -28.

$$x^{2}-3x-28=0$$

To solve the equation, factor x^{2}-3x-28 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.

$$a+b=-3 ab=-28$$

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 -28.

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

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=-7 b=4$$

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

$$\left(x-7\right)\left(x+4\right)$$

To find equation solutions, solve x-7=0 and x+4=0.

$$x=7 x=-4$$

Steps Using Factoring By Grouping

Subtract 2x from both sides.

$$x^{2}-x-20-2x=8$$

Combine -x and -2x to get -3x.

$$x^{2}-3x-20=8$$

Subtract 8 from both sides.

$$x^{2}-3x-20-8=0$$

Subtract 8 from -20 to get -28.

$$x^{2}-3x-28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\left(-28\right)=-28$$

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 -28.

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

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=-7 b=4$$

Rewrite x^{2}-3x-28 as \left(x^{2}-7x\right)+\left(4x-28\right).

$$\left(x^{2}-7x\right)+\left(4x-28\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-7\right)+4\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x+4\right)$$

To find equation solutions, solve x-7=0 and x+4=0.

$$x=7 x=-4$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$x^{2}-x-20-2x=8$$

Combine -x and -2x to get -3x.

$$x^{2}-3x-20=8$$

Subtract 8 from both sides.

$$x^{2}-3x-20-8=0$$

Subtract 8 from -20 to get -28.

$$x^{2}-3x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-28\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$x=\frac{-\left(-3\right)±\sqrt{9+112}}{2}$$

Add 9 to 112.

$$x=\frac{-\left(-3\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-\left(-3\right)±11}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±11}{2}$$

Now solve the equation x=\frac{3±11}{2} when ± is plus. Add 3 to 11.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{3±11}{2} when ± is minus. Subtract 11 from 3.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=7 x=-4$$

Steps for Completing the Square

Subtract 2x from both sides.

$$x^{2}-x-20-2x=8$$

Combine -x and -2x to get -3x.

$$x^{2}-3x-20=8$$

Add 20 to both sides.

$$x^{2}-3x=8+20$$

Add 8 and 20 to get 28.

$$x^{2}-3x=28$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=28+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=28+\frac{9}{4}$$

Add 28 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{121}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{11}{2} x-\frac{3}{2}=-\frac{11}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=7 x=-4$$