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

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

Solve for x

$$x=-4$$

$$x=-1$$

Steps Using the Quadratic Formula

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

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

Subtract 2 from both sides.

$$x^{2}+5x+6-2=0$$

Subtract 2 from 6 to get 4.

$$x^{2}+5x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 5.

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

Multiply -4 times 4.

$$x=\frac{-5±\sqrt{25-16}}{2}$$

Add 25 to -16.

$$x=\frac{-5±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-5±3}{2}$$

Now solve the equation x=\frac{-5±3}{2} when ± is plus. Add -5 to 3.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-5±3}{2} when ± is minus. Subtract 3 from -5.

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

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=-1 x=-4$$

Steps for Completing the Square

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

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

Subtract 6 from both sides.

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

Subtract 6 from 2 to get -4.

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

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

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

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-4+\frac{25}{4}$$

Add -4 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{9}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+\frac{5}{2}=\frac{3}{2} x+\frac{5}{2}=-\frac{3}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=-1 x=-4$$