Solve 5/x-2-3/x+6=4/x

$$\frac { 5 } { x - 2 } - \frac { 3 } { x + 6 } = \frac { 4 } { x }$$

Solve for x

$$x=-2$$

$$x=12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right)\left(x+6\right), the least common multiple of x-2,x+6,x.

$$x\left(x+6\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$\left(x^{2}+6x\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x^{2}+6x by 5.

$$5x^{2}+30x-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(x^{2}-2x\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(3x^{2}-6x\right)=\left(x-2\right)\left(x+6\right)\times 4$$

To find the opposite of 3x^{2}-6x, find the opposite of each term.

$$5x^{2}+30x-3x^{2}+6x=\left(x-2\right)\left(x+6\right)\times 4$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}+30x+6x=\left(x-2\right)\left(x+6\right)\times 4$$

Combine 30x and 6x to get 36x.

$$2x^{2}+36x=\left(x-2\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x-2 by x+6 and combine like terms.

$$2x^{2}+36x=\left(x^{2}+4x-12\right)\times 4$$

Use the distributive property to multiply x^{2}+4x-12 by 4.

$$2x^{2}+36x=4x^{2}+16x-48$$

Subtract 4x^{2} from both sides.

$$2x^{2}+36x-4x^{2}=16x-48$$

Combine 2x^{2} and -4x^{2} to get -2x^{2}.

$$-2x^{2}+36x=16x-48$$

Subtract 16x from both sides.

$$-2x^{2}+36x-16x=-48$$

Combine 36x and -16x to get 20x.

$$-2x^{2}+20x=-48$$

Add 48 to both sides.

$$-2x^{2}+20x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 20 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-2\right)\times 48}}{2\left(-2\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-2\right)\times 48}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-20±\sqrt{400+8\times 48}}{2\left(-2\right)}$$

Multiply 8 times 48.

$$x=\frac{-20±\sqrt{400+384}}{2\left(-2\right)}$$

Add 400 to 384.

$$x=\frac{-20±\sqrt{784}}{2\left(-2\right)}$$

Take the square root of 784.

$$x=\frac{-20±28}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-20±28}{-4}$$

Now solve the equation x=\frac{-20±28}{-4} when ± is plus. Add -20 to 28.

$$x=\frac{8}{-4}$$

Divide 8 by -4.

$$x=-2$$

Now solve the equation x=\frac{-20±28}{-4} when ± is minus. Subtract 28 from -20.

$$x=-\frac{48}{-4}$$

Divide -48 by -4.

$$x=12$$

The equation is now solved.

$$x=-2 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right)\left(x+6\right), the least common multiple of x-2,x+6,x.

$$x\left(x+6\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$\left(x^{2}+6x\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x^{2}+6x by 5.

$$5x^{2}+30x-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(x^{2}-2x\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(3x^{2}-6x\right)=\left(x-2\right)\left(x+6\right)\times 4$$

To find the opposite of 3x^{2}-6x, find the opposite of each term.

$$5x^{2}+30x-3x^{2}+6x=\left(x-2\right)\left(x+6\right)\times 4$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}+30x+6x=\left(x-2\right)\left(x+6\right)\times 4$$

Combine 30x and 6x to get 36x.

$$2x^{2}+36x=\left(x-2\right)\left(x+6\right)\times 4$$

Use the distributive property to multiply x-2 by x+6 and combine like terms.

$$2x^{2}+36x=\left(x^{2}+4x-12\right)\times 4$$

Use the distributive property to multiply x^{2}+4x-12 by 4.

$$2x^{2}+36x=4x^{2}+16x-48$$

Subtract 4x^{2} from both sides.

$$2x^{2}+36x-4x^{2}=16x-48$$

Combine 2x^{2} and -4x^{2} to get -2x^{2}.

$$-2x^{2}+36x=16x-48$$

Subtract 16x from both sides.

$$-2x^{2}+36x-16x=-48$$

Combine 36x and -16x to get 20x.

$$-2x^{2}+20x=-48$$

Divide both sides by -2.

$$\frac{-2x^{2}+20x}{-2}=-\frac{48}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{20}{-2}x=-\frac{48}{-2}$$

Divide 20 by -2.

$$x^{2}-10x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x^{2}-10x=24$$

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

$$x^{2}-10x+\left(-5\right)^{2}=24+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=24+25$$

Add 24 to 25.

$$x^{2}-10x+25=49$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-5=7 x-5=-7$$

Add 5 to both sides of the equation.

$$x=12 x=-2$$