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

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

Solve for x (complex solution)

$$x=-2+2i$$

$$x=-2-2i$$

Steps Using the Quadratic Formula

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

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

Subtract x from both sides.

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

Combine 5x and -x to get 4x.

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

Add 2 to both sides.

$$x^{2}+4x+6+2=0$$

Add 6 and 2 to get 8.

$$x^{2}+4x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 8}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-4±\sqrt{16-32}}{2}$$

Add 16 to -32.

$$x=\frac{-4±\sqrt{-16}}{2}$$

Take the square root of -16.

$$x=\frac{-4±4i}{2}$$

Now solve the equation x=\frac{-4±4i}{2} when ± is plus. Add -4 to 4i.

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

Divide -4+4i by 2.

$$x=-2+2i$$

Now solve the equation x=\frac{-4±4i}{2} when ± is minus. Subtract 4i from -4.

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

Divide -4-4i by 2.

$$x=-2-2i$$

The equation is now solved.

$$x=-2+2i x=-2-2i$$

Steps for Completing the Square

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

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

Subtract x from both sides.

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

Combine 5x and -x to get 4x.

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

Subtract 6 from both sides.

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

Subtract 6 from -2 to get -8.

$$x^{2}+4x=-8$$

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

$$x^{2}+4x+2^{2}=-8+2^{2}$$

Square 2.

$$x^{2}+4x+4=-8+4$$

Add -8 to 4.

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

Factor x^{2}+4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+2=2i x+2=-2i$$

Subtract 2 from both sides of the equation.

$$x=-2+2i x=-2-2i$$