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

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

Solve for x

$$x = \frac{\sqrt{1249} + 39}{8} \approx 9.292649262$$

$$x=\frac{39-\sqrt{1249}}{8}\approx 0.457350738$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by 10\left(x-2\right)\left(x+1\right), the least common multiple of x+1,2\left(x-2\right),5.

$$\left(10x-20\right)\times 2+\left(5x+5\right)\times 3=4\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 10x-20 by 2.

$$20x-40+\left(5x+5\right)\times 3=4\left(x-2\right)\left(x+1\right)$$

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

$$20x-40+15x+15=4\left(x-2\right)\left(x+1\right)$$

Combine 20x and 15x to get 35x.

$$35x-40+15=4\left(x-2\right)\left(x+1\right)$$

Add -40 and 15 to get -25.

$$35x-25=4\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 4 by x-2.

$$35x-25=\left(4x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 4x-8 by x+1 and combine like terms.

$$35x-25=4x^{2}-4x-8$$

Subtract 4x^{2} from both sides.

$$35x-25-4x^{2}=-4x-8$$

Add 4x to both sides.

$$35x-25-4x^{2}+4x=-8$$

Combine 35x and 4x to get 39x.

$$39x-25-4x^{2}=-8$$

Add 8 to both sides.

$$39x-25-4x^{2}+8=0$$

Add -25 and 8 to get -17.

$$39x-17-4x^{2}=0$$

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.

$$-4x^{2}+39x-17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 39 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-39±\sqrt{39^{2}-4\left(-4\right)\left(-17\right)}}{2\left(-4\right)}$$

Square 39.

$$x=\frac{-39±\sqrt{1521-4\left(-4\right)\left(-17\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-39±\sqrt{1521+16\left(-17\right)}}{2\left(-4\right)}$$

Multiply 16 times -17.

$$x=\frac{-39±\sqrt{1521-272}}{2\left(-4\right)}$$

Add 1521 to -272.

$$x=\frac{-39±\sqrt{1249}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-39±\sqrt{1249}}{-8}$$

Now solve the equation x=\frac{-39±\sqrt{1249}}{-8} when ± is plus. Add -39 to \sqrt{1249}.

$$x=\frac{\sqrt{1249}-39}{-8}$$

Divide -39+\sqrt{1249} by -8.

$$x=\frac{39-\sqrt{1249}}{8}$$

Now solve the equation x=\frac{-39±\sqrt{1249}}{-8} when ± is minus. Subtract \sqrt{1249} from -39.

$$x=\frac{-\sqrt{1249}-39}{-8}$$

Divide -39-\sqrt{1249} by -8.

$$x=\frac{\sqrt{1249}+39}{8}$$

The equation is now solved.

$$x=\frac{39-\sqrt{1249}}{8} x=\frac{\sqrt{1249}+39}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by 10\left(x-2\right)\left(x+1\right), the least common multiple of x+1,2\left(x-2\right),5.

$$\left(10x-20\right)\times 2+\left(5x+5\right)\times 3=4\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 10x-20 by 2.

$$20x-40+\left(5x+5\right)\times 3=4\left(x-2\right)\left(x+1\right)$$

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

$$20x-40+15x+15=4\left(x-2\right)\left(x+1\right)$$

Combine 20x and 15x to get 35x.

$$35x-40+15=4\left(x-2\right)\left(x+1\right)$$

Add -40 and 15 to get -25.

$$35x-25=4\left(x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 4 by x-2.

$$35x-25=\left(4x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 4x-8 by x+1 and combine like terms.

$$35x-25=4x^{2}-4x-8$$

Subtract 4x^{2} from both sides.

$$35x-25-4x^{2}=-4x-8$$

Add 4x to both sides.

$$35x-25-4x^{2}+4x=-8$$

Combine 35x and 4x to get 39x.

$$39x-25-4x^{2}=-8$$

Add 25 to both sides.

$$39x-4x^{2}=-8+25$$

Add -8 and 25 to get 17.

$$39x-4x^{2}=17$$

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.

$$-4x^{2}+39x=17$$

Divide both sides by -4.

$$\frac{-4x^{2}+39x}{-4}=\frac{17}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{39}{-4}x=\frac{17}{-4}$$

Divide 39 by -4.

$$x^{2}-\frac{39}{4}x=\frac{17}{-4}$$

Divide 17 by -4.

$$x^{2}-\frac{39}{4}x=-\frac{17}{4}$$

Divide -\frac{39}{4}, the coefficient of the x term, by 2 to get -\frac{39}{8}. Then add the square of -\frac{39}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{39}{4}x+\left(-\frac{39}{8}\right)^{2}=-\frac{17}{4}+\left(-\frac{39}{8}\right)^{2}$$

Square -\frac{39}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{39}{4}x+\frac{1521}{64}=-\frac{17}{4}+\frac{1521}{64}$$

Add -\frac{17}{4} to \frac{1521}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{39}{4}x+\frac{1521}{64}=\frac{1249}{64}$$

Factor x^{2}-\frac{39}{4}x+\frac{1521}{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}.

$$\left(x-\frac{39}{8}\right)^{2}=\frac{1249}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{8}\right)^{2}}=\sqrt{\frac{1249}{64}}$$

Simplify.

$$x-\frac{39}{8}=\frac{\sqrt{1249}}{8} x-\frac{39}{8}=-\frac{\sqrt{1249}}{8}$$

Add \frac{39}{8} to both sides of the equation.

$$x=\frac{\sqrt{1249}+39}{8} x=\frac{39-\sqrt{1249}}{8}$$