Solve 2x/x-8+3x/x+5=51/6

$$\frac{ 2x }{ x-8 } + \frac{ 3x }{ x+5 } = 5 \frac { 1 } { 6 }$$

Solve for x

$$x=-31$$

$$x=40$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,8 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-8\right)\left(x+5\right), the least common multiple of x-8,x+5,6.

$$\left(6x+30\right)\times 2x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 6x+30 by 2.

$$\left(12x+60\right)x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 12x+60 by x.

$$12x^{2}+60x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 6x-48 by 3.

$$12x^{2}+60x+\left(18x-144\right)x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 18x-144 by x.

$$12x^{2}+60x+18x^{2}-144x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Combine 12x^{2} and 18x^{2} to get 30x^{2}.

$$30x^{2}+60x-144x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Combine 60x and -144x to get -84x.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Multiply 5 and 6 to get 30.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\left(30+1\right)$$

Add 30 and 1 to get 31.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\times 31$$

Use the distributive property to multiply x-8 by x+5 and combine like terms.

$$30x^{2}-84x=\left(x^{2}-3x-40\right)\times 31$$

Use the distributive property to multiply x^{2}-3x-40 by 31.

$$30x^{2}-84x=31x^{2}-93x-1240$$

Subtract 31x^{2} from both sides.

$$30x^{2}-84x-31x^{2}=-93x-1240$$

Combine 30x^{2} and -31x^{2} to get -x^{2}.

$$-x^{2}-84x=-93x-1240$$

Add 93x to both sides.

$$-x^{2}-84x+93x=-1240$$

Combine -84x and 93x to get 9x.

$$-x^{2}+9x=-1240$$

Add 1240 to both sides.

$$-x^{2}+9x+1240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and 1240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 1240}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\times 1240}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\times 1240}}{2\left(-1\right)}$$

Multiply 4 times 1240.

$$x=\frac{-9±\sqrt{81+4960}}{2\left(-1\right)}$$

Add 81 to 4960.

$$x=\frac{-9±\sqrt{5041}}{2\left(-1\right)}$$

Take the square root of 5041.

$$x=\frac{-9±71}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±71}{-2}$$

Now solve the equation x=\frac{-9±71}{-2} when ± is plus. Add -9 to 71.

$$x=\frac{62}{-2}$$

Divide 62 by -2.

$$x=-31$$

Now solve the equation x=\frac{-9±71}{-2} when ± is minus. Subtract 71 from -9.

$$x=-\frac{80}{-2}$$

Divide -80 by -2.

$$x=40$$

The equation is now solved.

$$x=-31 x=40$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,8 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-8\right)\left(x+5\right), the least common multiple of x-8,x+5,6.

$$\left(6x+30\right)\times 2x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 6x+30 by 2.

$$\left(12x+60\right)x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 12x+60 by x.

$$12x^{2}+60x+\left(6x-48\right)\times 3x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 6x-48 by 3.

$$12x^{2}+60x+\left(18x-144\right)x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Use the distributive property to multiply 18x-144 by x.

$$12x^{2}+60x+18x^{2}-144x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Combine 12x^{2} and 18x^{2} to get 30x^{2}.

$$30x^{2}+60x-144x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Combine 60x and -144x to get -84x.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)$$

Multiply 5 and 6 to get 30.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\left(30+1\right)$$

Add 30 and 1 to get 31.

$$30x^{2}-84x=\left(x-8\right)\left(x+5\right)\times 31$$

Use the distributive property to multiply x-8 by x+5 and combine like terms.

$$30x^{2}-84x=\left(x^{2}-3x-40\right)\times 31$$

Use the distributive property to multiply x^{2}-3x-40 by 31.

$$30x^{2}-84x=31x^{2}-93x-1240$$

Subtract 31x^{2} from both sides.

$$30x^{2}-84x-31x^{2}=-93x-1240$$

Combine 30x^{2} and -31x^{2} to get -x^{2}.

$$-x^{2}-84x=-93x-1240$$

Add 93x to both sides.

$$-x^{2}-84x+93x=-1240$$

Combine -84x and 93x to get 9x.

$$-x^{2}+9x=-1240$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=-\frac{1240}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=-\frac{1240}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=-\frac{1240}{-1}$$

Divide -1240 by -1.

$$x^{2}-9x=1240$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=1240+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=1240+\frac{81}{4}$$

Add 1240 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{5041}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{5041}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{5041}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{71}{2} x-\frac{9}{2}=-\frac{71}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=40 x=-31$$