Solve 5x/3(x-1)+6/x+2=5/2

$$\frac { 5 x } { 3 ( x - 1 ) } + \frac { 6 } { x + 2 } = \frac { 5 } { 2 }$$

Solve for x

$$x = \frac{\sqrt{1561} + 41}{10} \approx 8.050949253$$

$$x=\frac{41-\sqrt{1561}}{10}\approx 0.149050747$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+2\right), the least common multiple of 3\left(x-1\right),x+2,2.

$$\left(2x+4\right)\times 5x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 5.

$$\left(10x+20\right)x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 10x+20 by x.

$$10x^{2}+20x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-6 by 6.

$$10x^{2}+20x+36x-36=15\left(x-1\right)\left(x+2\right)$$

Combine 20x and 36x to get 56x.

$$10x^{2}+56x-36=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 15 by x-1.

$$10x^{2}+56x-36=\left(15x-15\right)\left(x+2\right)$$

Use the distributive property to multiply 15x-15 by x+2 and combine like terms.

$$10x^{2}+56x-36=15x^{2}+15x-30$$

Subtract 15x^{2} from both sides.

$$10x^{2}+56x-36-15x^{2}=15x-30$$

Combine 10x^{2} and -15x^{2} to get -5x^{2}.

$$-5x^{2}+56x-36=15x-30$$

Subtract 15x from both sides.

$$-5x^{2}+56x-36-15x=-30$$

Combine 56x and -15x to get 41x.

$$-5x^{2}+41x-36=-30$$

Add 30 to both sides.

$$-5x^{2}+41x-36+30=0$$

Add -36 and 30 to get -6.

$$-5x^{2}+41x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 41 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-41±\sqrt{41^{2}-4\left(-5\right)\left(-6\right)}}{2\left(-5\right)}$$

Square 41.

$$x=\frac{-41±\sqrt{1681-4\left(-5\right)\left(-6\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-41±\sqrt{1681+20\left(-6\right)}}{2\left(-5\right)}$$

Multiply 20 times -6.

$$x=\frac{-41±\sqrt{1681-120}}{2\left(-5\right)}$$

Add 1681 to -120.

$$x=\frac{-41±\sqrt{1561}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-41±\sqrt{1561}}{-10}$$

Now solve the equation x=\frac{-41±\sqrt{1561}}{-10} when ± is plus. Add -41 to \sqrt{1561}.

$$x=\frac{\sqrt{1561}-41}{-10}$$

Divide -41+\sqrt{1561} by -10.

$$x=\frac{41-\sqrt{1561}}{10}$$

Now solve the equation x=\frac{-41±\sqrt{1561}}{-10} when ± is minus. Subtract \sqrt{1561} from -41.

$$x=\frac{-\sqrt{1561}-41}{-10}$$

Divide -41-\sqrt{1561} by -10.

$$x=\frac{\sqrt{1561}+41}{10}$$

The equation is now solved.

$$x=\frac{41-\sqrt{1561}}{10} x=\frac{\sqrt{1561}+41}{10}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-1\right)\left(x+2\right), the least common multiple of 3\left(x-1\right),x+2,2.

$$\left(2x+4\right)\times 5x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 5.

$$\left(10x+20\right)x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 10x+20 by x.

$$10x^{2}+20x+\left(6x-6\right)\times 6=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-6 by 6.

$$10x^{2}+20x+36x-36=15\left(x-1\right)\left(x+2\right)$$

Combine 20x and 36x to get 56x.

$$10x^{2}+56x-36=15\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 15 by x-1.

$$10x^{2}+56x-36=\left(15x-15\right)\left(x+2\right)$$

Use the distributive property to multiply 15x-15 by x+2 and combine like terms.

$$10x^{2}+56x-36=15x^{2}+15x-30$$

Subtract 15x^{2} from both sides.

$$10x^{2}+56x-36-15x^{2}=15x-30$$

Combine 10x^{2} and -15x^{2} to get -5x^{2}.

$$-5x^{2}+56x-36=15x-30$$

Subtract 15x from both sides.

$$-5x^{2}+56x-36-15x=-30$$

Combine 56x and -15x to get 41x.

$$-5x^{2}+41x-36=-30$$

Add 36 to both sides.

$$-5x^{2}+41x=-30+36$$

Add -30 and 36 to get 6.

$$-5x^{2}+41x=6$$

Divide both sides by -5.

$$\frac{-5x^{2}+41x}{-5}=\frac{6}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{41}{-5}x=\frac{6}{-5}$$

Divide 41 by -5.

$$x^{2}-\frac{41}{5}x=\frac{6}{-5}$$

Divide 6 by -5.

$$x^{2}-\frac{41}{5}x=-\frac{6}{5}$$

Divide -\frac{41}{5}, the coefficient of the x term, by 2 to get -\frac{41}{10}. Then add the square of -\frac{41}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{41}{5}x+\left(-\frac{41}{10}\right)^{2}=-\frac{6}{5}+\left(-\frac{41}{10}\right)^{2}$$

Square -\frac{41}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{41}{5}x+\frac{1681}{100}=-\frac{6}{5}+\frac{1681}{100}$$

Add -\frac{6}{5} to \frac{1681}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{41}{5}x+\frac{1681}{100}=\frac{1561}{100}$$

Factor x^{2}-\frac{41}{5}x+\frac{1681}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{41}{10}\right)^{2}=\frac{1561}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{10}\right)^{2}}=\sqrt{\frac{1561}{100}}$$

Simplify.

$$x-\frac{41}{10}=\frac{\sqrt{1561}}{10} x-\frac{41}{10}=-\frac{\sqrt{1561}}{10}$$

Add \frac{41}{10} to both sides of the equation.

$$x=\frac{\sqrt{1561}+41}{10} x=\frac{41-\sqrt{1561}}{10}$$