Solve x-3/x+2=5/x+2

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

Solve for x

$$x=-4$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\left(x+2\right)x-3=5$$

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

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

Subtract 5 from both sides.

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

Subtract 5 from -3 to get -8.

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

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

$$x=\frac{-2±\sqrt{2^{2}-4\left(-8\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$x=\frac{-2±\sqrt{4+32}}{2}$$

Add 4 to 32.

$$x=\frac{-2±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-2±6}{2}$$

Now solve the equation x=\frac{-2±6}{2} when ± is plus. Add -2 to 6.

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

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-2±6}{2} when ± is minus. Subtract 6 from -2.

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

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=2 x=-4$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\left(x+2\right)x-3=5$$

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

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

Add 3 to both sides.

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

Add 5 and 3 to get 8.

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

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.

$$x^{2}+2x+1^{2}=8+1^{2}$$

Square 1.

$$x^{2}+2x+1=8+1$$

Add 8 to 1.

$$x^{2}+2x+1=9$$

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}.

$$\left(x+1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+1=3 x+1=-3$$

Subtract 1 from both sides of the equation.

$$x=2 x=-4$$