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

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

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x+2.

$$3x^{2}+6x-\left(x+1\right)\left(x-2\right)=2$$

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

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

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

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

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

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

Combine 6x and x to get 7x.

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

Subtract 2 from both sides.

$$2x^{2}+7x+2-2=0$$

Subtract 2 from 2 to get 0.

$$2x^{2}+7x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}}}{2\times 2}$$

Take the square root of 7^{2}.

$$x=\frac{-7±7}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-7±7}{4}$$

Now solve the equation x=\frac{-7±7}{4} when ± is plus. Add -7 to 7.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

Now solve the equation x=\frac{-7±7}{4} when ± is minus. Subtract 7 from -7.

$$x=-\frac{14}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=0 x=-\frac{7}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x+2.

$$3x^{2}+6x-\left(x+1\right)\left(x-2\right)=2$$

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

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

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

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

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

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

Combine 6x and x to get 7x.

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

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

$$2x^{2}+7x=0$$

Divide both sides by 2.

$$\frac{2x^{2}+7x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{7}{2}x=\frac{0}{2}$$

Divide 0 by 2.

$$x^{2}+\frac{7}{2}x=0$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{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}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{7}{4} x+\frac{7}{4}=-\frac{7}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=0 x=-\frac{7}{2}$$