Solve (x-6)(x+1)-(2-x)(x+3)=36

$$( x - 6 ) ( x + 1 ) - ( 2 - x ) ( x + 3 ) = 36$$

Solve for x

$$x=-4$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-6 by x+1 and combine like terms.

$$x^{2}-5x-6-\left(2-x\right)\left(x+3\right)=36$$

Use the distributive property to multiply 2-x by x+3 and combine like terms.

$$x^{2}-5x-6-\left(-x+6-x^{2}\right)=36$$

To find the opposite of -x+6-x^{2}, find the opposite of each term.

$$x^{2}-5x-6+x-6+x^{2}=36$$

Combine -5x and x to get -4x.

$$x^{2}-4x-6-6+x^{2}=36$$

Subtract 6 from -6 to get -12.

$$x^{2}-4x-12+x^{2}=36$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-4x-12=36$$

Subtract 36 from both sides.

$$2x^{2}-4x-12-36=0$$

Subtract 36 from -12 to get -48.

$$2x^{2}-4x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-48\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-48\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-48\right)}}{2\times 2}$$

Multiply -8 times -48.

$$x=\frac{-\left(-4\right)±\sqrt{16+384}}{2\times 2}$$

Add 16 to 384.

$$x=\frac{-\left(-4\right)±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-\left(-4\right)±20}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±20}{4}$$

Now solve the equation x=\frac{4±20}{4} when ± is plus. Add 4 to 20.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{4±20}{4} when ± is minus. Subtract 20 from 4.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

$$x=6 x=-4$$

Steps for Completing the Square

Use the distributive property to multiply x-6 by x+1 and combine like terms.

$$x^{2}-5x-6-\left(2-x\right)\left(x+3\right)=36$$

Use the distributive property to multiply 2-x by x+3 and combine like terms.

$$x^{2}-5x-6-\left(-x+6-x^{2}\right)=36$$

To find the opposite of -x+6-x^{2}, find the opposite of each term.

$$x^{2}-5x-6+x-6+x^{2}=36$$

Combine -5x and x to get -4x.

$$x^{2}-4x-6-6+x^{2}=36$$

Subtract 6 from -6 to get -12.

$$x^{2}-4x-12+x^{2}=36$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-4x-12=36$$

Add 12 to both sides.

$$2x^{2}-4x=36+12$$

Add 36 and 12 to get 48.

$$2x^{2}-4x=48$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{48}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{48}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{48}{2}$$

Divide 48 by 2.

$$x^{2}-2x=24$$

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.

$$x^{2}-2x+1=24+1$$

Add 24 to 1.

$$x^{2}-2x+1=25$$

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}.

$$\left(x-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-1=5 x-1=-5$$

Add 1 to both sides of the equation.

$$x=6 x=-4$$