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

$$\frac { x } { x + 3 } = \frac { 3 } { ( x + 2 ) ( x + 3 ) }$$

Solve for x

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x+3,\left(x+2\right)\left(x+3\right).

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

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

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

Subtract 3 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)}}{2}$$

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12}}{2}$$

Add 4 to 12.

$$x=\frac{-2±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-2±4}{2}$$

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

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

Divide 2 by 2.

$$x=1$$

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

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=1 x=-3$$

Variable x cannot be equal to -3.

$$x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x+3,\left(x+2\right)\left(x+3\right).

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

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

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

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.

$$x^{2}+2x+1^{2}=3+1^{2}$$

Square 1.

$$x^{2}+2x+1=3+1$$

Add 3 to 1.

$$x^{2}+2x+1=4$$

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}.

$$\left(x+1\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{4}$$

Simplify.

$$x+1=2 x+1=-2$$

Subtract 1 from both sides of the equation.

$$x=1 x=-3$$

Variable x cannot be equal to -3.

$$x=1$$