Solve 6x/x+4+4=2x+2/x-1

$$\frac { 6 x } { x + 4 } + 4 = \frac { 2 x + 2 } { x - 1 }$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+4\right), the least common multiple of x+4,x-1.

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

Use the distributive property to multiply x-1 by 6.

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

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

$$6x^{2}-6x+\left(x-1\right)\left(x+4\right)\times 4=\left(x+4\right)\left(2x+2\right)$$

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

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

Use the distributive property to multiply x^{2}+3x-4 by 4.

$$6x^{2}-6x+4x^{2}+12x-16=\left(x+4\right)\left(2x+2\right)$$

Combine 6x^{2} and 4x^{2} to get 10x^{2}.

$$10x^{2}-6x+12x-16=\left(x+4\right)\left(2x+2\right)$$

Combine -6x and 12x to get 6x.

$$10x^{2}+6x-16=\left(x+4\right)\left(2x+2\right)$$

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

$$10x^{2}+6x-16=2x^{2}+10x+8$$

Subtract 2x^{2} from both sides.

$$10x^{2}+6x-16-2x^{2}=10x+8$$

Combine 10x^{2} and -2x^{2} to get 8x^{2}.

$$8x^{2}+6x-16=10x+8$$

Subtract 10x from both sides.

$$8x^{2}+6x-16-10x=8$$

Combine 6x and -10x to get -4x.

$$8x^{2}-4x-16=8$$

Subtract 8 from both sides.

$$8x^{2}-4x-16-8=0$$

Subtract 8 from -16 to get -24.

$$8x^{2}-4x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -4 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 8\left(-24\right)}}{2\times 8}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 8\left(-24\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-4\right)±\sqrt{16-32\left(-24\right)}}{2\times 8}$$

Multiply -32 times -24.

$$x=\frac{-\left(-4\right)±\sqrt{16+768}}{2\times 8}$$

Add 16 to 768.

$$x=\frac{-\left(-4\right)±\sqrt{784}}{2\times 8}$$

Take the square root of 784.

$$x=\frac{-\left(-4\right)±28}{2\times 8}$$

The opposite of -4 is 4.

$$x=\frac{4±28}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{4±28}{16}$$

Now solve the equation x=\frac{4±28}{16} when ± is plus. Add 4 to 28.

$$x=\frac{32}{16}$$

Divide 32 by 16.

$$x=2$$

Now solve the equation x=\frac{4±28}{16} when ± is minus. Subtract 28 from 4.

$$x=-\frac{24}{16}$$

Reduce the fraction \frac{-24}{16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=2 x=-\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+4\right), the least common multiple of x+4,x-1.

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

Use the distributive property to multiply x-1 by 6.

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

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

$$6x^{2}-6x+\left(x-1\right)\left(x+4\right)\times 4=\left(x+4\right)\left(2x+2\right)$$

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

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

Use the distributive property to multiply x^{2}+3x-4 by 4.

$$6x^{2}-6x+4x^{2}+12x-16=\left(x+4\right)\left(2x+2\right)$$

Combine 6x^{2} and 4x^{2} to get 10x^{2}.

$$10x^{2}-6x+12x-16=\left(x+4\right)\left(2x+2\right)$$

Combine -6x and 12x to get 6x.

$$10x^{2}+6x-16=\left(x+4\right)\left(2x+2\right)$$

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

$$10x^{2}+6x-16=2x^{2}+10x+8$$

Subtract 2x^{2} from both sides.

$$10x^{2}+6x-16-2x^{2}=10x+8$$

Combine 10x^{2} and -2x^{2} to get 8x^{2}.

$$8x^{2}+6x-16=10x+8$$

Subtract 10x from both sides.

$$8x^{2}+6x-16-10x=8$$

Combine 6x and -10x to get -4x.

$$8x^{2}-4x-16=8$$

Add 16 to both sides.

$$8x^{2}-4x=8+16$$

Add 8 and 16 to get 24.

$$8x^{2}-4x=24$$

Divide both sides by 8.

$$\frac{8x^{2}-4x}{8}=\frac{24}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{4}{8}\right)x=\frac{24}{8}$$

Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{1}{2}x=\frac{24}{8}$$

Divide 24 by 8.

$$x^{2}-\frac{1}{2}x=3$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=3+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=3+\frac{1}{16}$$

Add 3 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{7}{4} x-\frac{1}{4}=-\frac{7}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=2 x=-\frac{3}{2}$$