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

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

Solve for x

$$x = \frac{\sqrt{37} + 3}{2} \approx 4.541381265$$

$$x=\frac{3-\sqrt{37}}{2}\approx -1.541381265$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{x+2}{x+2}.

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

Since \frac{5\left(x+2\right)}{x+2} and \frac{3}{x+2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\left(x+2\right)-3.

$$x=\frac{5x+10-3}{x+2}$$

Combine like terms in 5x+10-3.

$$x=\frac{5x+7}{x+2}$$

Subtract \frac{5x+7}{x+2} from both sides.

$$x-\frac{5x+7}{x+2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+2}{x+2}.

$$\frac{x\left(x+2\right)}{x+2}-\frac{5x+7}{x+2}=0$$

Since \frac{x\left(x+2\right)}{x+2} and \frac{5x+7}{x+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+2\right)-\left(5x+7\right)}{x+2}=0$$

Do the multiplications in x\left(x+2\right)-\left(5x+7\right).

$$\frac{x^{2}+2x-5x-7}{x+2}=0$$

Combine like terms in x^{2}+2x-5x-7.

$$\frac{x^{2}-3x-7}{x+2}=0$$

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

$$x^{2}-3x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-7\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-\left(-3\right)±\sqrt{9+28}}{2}$$

Add 9 to 28.

$$x=\frac{-\left(-3\right)±\sqrt{37}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{37}}{2}$$

Now solve the equation x=\frac{3±\sqrt{37}}{2} when ± is plus. Add 3 to \sqrt{37}.

$$x=\frac{\sqrt{37}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from 3.

$$x=\frac{3-\sqrt{37}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+3}{2} x=\frac{3-\sqrt{37}}{2}$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{x+2}{x+2}.

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

Since \frac{5\left(x+2\right)}{x+2} and \frac{3}{x+2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\left(x+2\right)-3.

$$x=\frac{5x+10-3}{x+2}$$

Combine like terms in 5x+10-3.

$$x=\frac{5x+7}{x+2}$$

Subtract \frac{5x+7}{x+2} from both sides.

$$x-\frac{5x+7}{x+2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+2}{x+2}.

$$\frac{x\left(x+2\right)}{x+2}-\frac{5x+7}{x+2}=0$$

Since \frac{x\left(x+2\right)}{x+2} and \frac{5x+7}{x+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+2\right)-\left(5x+7\right)}{x+2}=0$$

Do the multiplications in x\left(x+2\right)-\left(5x+7\right).

$$\frac{x^{2}+2x-5x-7}{x+2}=0$$

Combine like terms in x^{2}+2x-5x-7.

$$\frac{x^{2}-3x-7}{x+2}=0$$

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

$$x^{2}-3x-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$x^{2}-3x=7$$

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.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=7+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=7+\frac{9}{4}$$

Add 7 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{37}{4}$$

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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{37}}{2} x-\frac{3}{2}=-\frac{\sqrt{37}}{2}$$

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

$$x=\frac{\sqrt{37}+3}{2} x=\frac{3-\sqrt{37}}{2}$$