Solve (2x+3)(x-2)-x(x+1)=0

$$( 2 x + 3 ) ( x - 2 ) - x ( x + 1 ) = 0$$

Solve for x

$$x=\sqrt{7}+1\approx 3.645751311$$

$$x=1-\sqrt{7}\approx -1.645751311$$

Steps Using the Quadratic Formula

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

$$2x^{2}-x-6-x\left(x+1\right)=0$$

Use the distributive property to multiply x by x+1.

$$2x^{2}-x-6-\left(x^{2}+x\right)=0$$

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

$$2x^{2}-x-6-x^{2}-x=0$$

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

$$x^{2}-x-6-x=0$$

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

$$x^{2}-2x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-6\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-\left(-2\right)±\sqrt{4+24}}{2}$$

Add 4 to 24.

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

Take the square root of 28.

$$x=\frac{-\left(-2\right)±2\sqrt{7}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{7}}{2}$$

Now solve the equation x=\frac{2±2\sqrt{7}}{2} when ± is plus. Add 2 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+2}{2}$$

Divide 2+2\sqrt{7} by 2.

$$x=\sqrt{7}+1$$

Now solve the equation x=\frac{2±2\sqrt{7}}{2} when ± is minus. Subtract 2\sqrt{7} from 2.

$$x=\frac{2-2\sqrt{7}}{2}$$

Divide 2-2\sqrt{7} by 2.

$$x=1-\sqrt{7}$$

The equation is now solved.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$

Steps for Completing the Square

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

$$2x^{2}-x-6-x\left(x+1\right)=0$$

Use the distributive property to multiply x by x+1.

$$2x^{2}-x-6-\left(x^{2}+x\right)=0$$

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

$$2x^{2}-x-6-x^{2}-x=0$$

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

$$x^{2}-x-6-x=0$$

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

$$x^{2}-2x-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

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

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.

$$x^{2}-2x+1=6+1$$

Add 6 to 1.

$$x^{2}-2x+1=7$$

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}.

$$\left(x-1\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-1=\sqrt{7} x-1=-\sqrt{7}$$

Add 1 to both sides of the equation.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$