Solve 3/x+2+5/2x-1=2

$$\frac{ 3 }{ x+2 } + \frac{ 5 }{ 2x-1 } = 2$$

Solve for x

$$x = \frac{\sqrt{201} + 5}{8} \approx 2.39718086$$

$$x=\frac{5-\sqrt{201}}{8}\approx -1.14718086$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(x+2\right), the least common multiple of x+2,2x-1.

$$\left(2x-1\right)\times 3+\left(x+2\right)\times 5=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2x-1 by 3.

$$6x-3+\left(x+2\right)\times 5=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 5.

$$6x-3+5x+10=2\left(2x-1\right)\left(x+2\right)$$

Combine 6x and 5x to get 11x.

$$11x-3+10=2\left(2x-1\right)\left(x+2\right)$$

Add -3 and 10 to get 7.

$$11x+7=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2 by 2x-1.

$$11x+7=\left(4x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 4x-2 by x+2 and combine like terms.

$$11x+7=4x^{2}+6x-4$$

Subtract 4x^{2} from both sides.

$$11x+7-4x^{2}=6x-4$$

Subtract 6x from both sides.

$$11x+7-4x^{2}-6x=-4$$

Combine 11x and -6x to get 5x.

$$5x+7-4x^{2}=-4$$

Add 4 to both sides.

$$5x+7-4x^{2}+4=0$$

Add 7 and 4 to get 11.

$$5x+11-4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}+5x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 11}}{2\left(-4\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-4\right)\times 11}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-5±\sqrt{25+16\times 11}}{2\left(-4\right)}$$

Multiply 16 times 11.

$$x=\frac{-5±\sqrt{25+176}}{2\left(-4\right)}$$

Add 25 to 176.

$$x=\frac{-5±\sqrt{201}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-5±\sqrt{201}}{-8}$$

Now solve the equation x=\frac{-5±\sqrt{201}}{-8} when ± is plus. Add -5 to \sqrt{201}.

$$x=\frac{\sqrt{201}-5}{-8}$$

Divide -5+\sqrt{201} by -8.

$$x=\frac{5-\sqrt{201}}{8}$$

Now solve the equation x=\frac{-5±\sqrt{201}}{-8} when ± is minus. Subtract \sqrt{201} from -5.

$$x=\frac{-\sqrt{201}-5}{-8}$$

Divide -5-\sqrt{201} by -8.

$$x=\frac{\sqrt{201}+5}{8}$$

The equation is now solved.

$$x=\frac{5-\sqrt{201}}{8} x=\frac{\sqrt{201}+5}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(x+2\right), the least common multiple of x+2,2x-1.

$$\left(2x-1\right)\times 3+\left(x+2\right)\times 5=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2x-1 by 3.

$$6x-3+\left(x+2\right)\times 5=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 5.

$$6x-3+5x+10=2\left(2x-1\right)\left(x+2\right)$$

Combine 6x and 5x to get 11x.

$$11x-3+10=2\left(2x-1\right)\left(x+2\right)$$

Add -3 and 10 to get 7.

$$11x+7=2\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2 by 2x-1.

$$11x+7=\left(4x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 4x-2 by x+2 and combine like terms.

$$11x+7=4x^{2}+6x-4$$

Subtract 4x^{2} from both sides.

$$11x+7-4x^{2}=6x-4$$

Subtract 6x from both sides.

$$11x+7-4x^{2}-6x=-4$$

Combine 11x and -6x to get 5x.

$$5x+7-4x^{2}=-4$$

Subtract 7 from both sides.

$$5x-4x^{2}=-4-7$$

Subtract 7 from -4 to get -11.

$$5x-4x^{2}=-11$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-4x^{2}+5x=-11$$

Divide both sides by -4.

$$\frac{-4x^{2}+5x}{-4}=-\frac{11}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{5}{-4}x=-\frac{11}{-4}$$

Divide 5 by -4.

$$x^{2}-\frac{5}{4}x=-\frac{11}{-4}$$

Divide -11 by -4.

$$x^{2}-\frac{5}{4}x=\frac{11}{4}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=\frac{11}{4}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{11}{4}+\frac{25}{64}$$

Add \frac{11}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{201}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{201}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{201}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{\sqrt{201}}{8} x-\frac{5}{8}=-\frac{\sqrt{201}}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{\sqrt{201}+5}{8} x=\frac{5-\sqrt{201}}{8}$$