Solve (2)(x+2)(2x-1)-(x-2)(x+3)=x^2

$$( 2 ) ( x + 2 ) ( 2 x - 1 ) - ( x - 2 ) ( x + 3 ) = x ^ { 2 }$$

Solve for x

$$x=-2$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 2 by x+2.

$$\left(2x+4\right)\left(2x-1\right)-\left(x-2\right)\left(x+3\right)=x^{2}$$

Use the distributive property to multiply 2x+4 by 2x-1 and combine like terms.

$$4x^{2}+6x-4-\left(x-2\right)\left(x+3\right)=x^{2}$$

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

$$4x^{2}+6x-4-\left(x^{2}+x-6\right)=x^{2}$$

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

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

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+6x-4-x+6=x^{2}$$

Combine 6x and -x to get 5x.

$$3x^{2}+5x-4+6=x^{2}$$

Add -4 and 6 to get 2.

$$3x^{2}+5x+2=x^{2}$$

Subtract x^{2} from both sides.

$$3x^{2}+5x+2-x^{2}=0$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+5x+2=0$$

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

$$a+b=5 ab=2\times 2=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite 2x^{2}+5x+2 as \left(2x^{2}+x\right)+\left(4x+2\right).

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

Factor out x in the first and 2 in the second group.

$$x\left(2x+1\right)+2\left(2x+1\right)$$

Factor out common term 2x+1 by using distributive property.

$$\left(2x+1\right)\left(x+2\right)$$

To find equation solutions, solve 2x+1=0 and x+2=0.

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

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by x+2.

$$\left(2x+4\right)\left(2x-1\right)-\left(x-2\right)\left(x+3\right)=x^{2}$$

Use the distributive property to multiply 2x+4 by 2x-1 and combine like terms.

$$4x^{2}+6x-4-\left(x-2\right)\left(x+3\right)=x^{2}$$

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

$$4x^{2}+6x-4-\left(x^{2}+x-6\right)=x^{2}$$

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

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

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+6x-4-x+6=x^{2}$$

Combine 6x and -x to get 5x.

$$3x^{2}+5x-4+6=x^{2}$$

Add -4 and 6 to get 2.

$$3x^{2}+5x+2=x^{2}$$

Subtract x^{2} from both sides.

$$3x^{2}+5x+2-x^{2}=0$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+5x+2=0$$

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

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\times 2}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\times 2}}{2\times 2}$$

Multiply -8 times 2.

$$x=\frac{-5±\sqrt{25-16}}{2\times 2}$$

Add 25 to -16.

$$x=\frac{-5±\sqrt{9}}{2\times 2}$$

Take the square root of 9.

$$x=\frac{-5±3}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±3}{4}$$

Now solve the equation x=\frac{-5±3}{4} when ± is plus. Add -5 to 3.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-5±3}{4} when ± is minus. Subtract 3 from -5.

$$x=-\frac{8}{4}$$

Divide -8 by 4.

$$x=-2$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 2 by x+2.

$$\left(2x+4\right)\left(2x-1\right)-\left(x-2\right)\left(x+3\right)=x^{2}$$

Use the distributive property to multiply 2x+4 by 2x-1 and combine like terms.

$$4x^{2}+6x-4-\left(x-2\right)\left(x+3\right)=x^{2}$$

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

$$4x^{2}+6x-4-\left(x^{2}+x-6\right)=x^{2}$$

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

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

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+6x-4-x+6=x^{2}$$

Combine 6x and -x to get 5x.

$$3x^{2}+5x-4+6=x^{2}$$

Add -4 and 6 to get 2.

$$3x^{2}+5x+2=x^{2}$$

Subtract x^{2} from both sides.

$$3x^{2}+5x+2-x^{2}=0$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+5x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by 2.

$$\frac{2x^{2}+5x}{2}=-\frac{2}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5}{2}x=-\frac{2}{2}$$

Divide -2 by 2.

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

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=-1+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=-1+\frac{25}{16}$$

Add -1 to \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{9}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{3}{4} x+\frac{5}{4}=-\frac{3}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

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