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

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

Solve for x

$$x = \frac{\sqrt{69} + 3}{2} \approx 5.653311931$$

$$x=\frac{3-\sqrt{69}}{2}\approx -2.653311931$$

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 5\left(x+2\right), the least common multiple of 5,x+2.

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

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

$$x^{2}+2x=5\left(x+3\right)$$

Use the distributive property to multiply 5 by x+3.

$$x^{2}+2x=5x+15$$

Subtract 5x from both sides.

$$x^{2}+2x-5x=15$$

Combine 2x and -5x to get -3x.

$$x^{2}-3x=15$$

Subtract 15 from both sides.

$$x^{2}-3x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-15\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-\left(-3\right)±\sqrt{9+60}}{2}$$

Add 9 to 60.

$$x=\frac{-\left(-3\right)±\sqrt{69}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{69}}{2}$$

Now solve the equation x=\frac{3±\sqrt{69}}{2} when ± is plus. Add 3 to \sqrt{69}.

$$x=\frac{\sqrt{69}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{69}}{2} when ± is minus. Subtract \sqrt{69} from 3.

$$x=\frac{3-\sqrt{69}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{69}+3}{2} x=\frac{3-\sqrt{69}}{2}$$

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 5\left(x+2\right), the least common multiple of 5,x+2.

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

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

$$x^{2}+2x=5\left(x+3\right)$$

Use the distributive property to multiply 5 by x+3.

$$x^{2}+2x=5x+15$$

Subtract 5x from both sides.

$$x^{2}+2x-5x=15$$

Combine 2x and -5x to get -3x.

$$x^{2}-3x=15$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=15+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=15+\frac{9}{4}$$

Add 15 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{69}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{69}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{69}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{69}}{2} x-\frac{3}{2}=-\frac{\sqrt{69}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{69}+3}{2} x=\frac{3-\sqrt{69}}{2}$$