Solve 5=1/x+1+4x-2/x+4

$$5= \frac{ 1 }{ x+1 } + \frac{ 4x-2 }{ x+4 }$$

Solve for x (complex solution)

$$x=\sqrt{103}-11\approx -0.851108435$$

$$x=-\left(\sqrt{103}+11\right)\approx -21.148891565$$

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+1,x+4.

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

Use the distributive property to multiply 5 by x+1.

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

Use the distributive property to multiply 5x+5 by x+4 and combine like terms.

$$5x^{2}+25x+20=x+4+\left(x+1\right)\left(4x-2\right)$$

Use the distributive property to multiply x+1 by 4x-2 and combine like terms.

$$5x^{2}+25x+20=x+4+4x^{2}+2x-2$$

Combine x and 2x to get 3x.

$$5x^{2}+25x+20=3x+4+4x^{2}-2$$

Subtract 2 from 4 to get 2.

$$5x^{2}+25x+20=3x+2+4x^{2}$$

Subtract 3x from both sides.

$$5x^{2}+25x+20-3x=2+4x^{2}$$

Combine 25x and -3x to get 22x.

$$5x^{2}+22x+20=2+4x^{2}$$

Subtract 2 from both sides.

$$5x^{2}+22x+20-2=4x^{2}$$

Subtract 2 from 20 to get 18.

$$5x^{2}+22x+18=4x^{2}$$

Subtract 4x^{2} from both sides.

$$5x^{2}+22x+18-4x^{2}=0$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+22x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 18}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-22±\sqrt{484-72}}{2}$$

Add 484 to -72.

$$x=\frac{-22±\sqrt{412}}{2}$$

Take the square root of 412.

$$x=\frac{-22±2\sqrt{103}}{2}$$

Now solve the equation x=\frac{-22±2\sqrt{103}}{2} when ± is plus. Add -22 to 2\sqrt{103}.

$$x=\frac{2\sqrt{103}-22}{2}$$

Divide -22+2\sqrt{103} by 2.

$$x=\sqrt{103}-11$$

Now solve the equation x=\frac{-22±2\sqrt{103}}{2} when ± is minus. Subtract 2\sqrt{103} from -22.

$$x=\frac{-2\sqrt{103}-22}{2}$$

Divide -22-2\sqrt{103} by 2.

$$x=-\sqrt{103}-11$$

The equation is now solved.

$$x=\sqrt{103}-11 x=-\sqrt{103}-11$$

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+1,x+4.

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

Use the distributive property to multiply 5 by x+1.

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

Use the distributive property to multiply 5x+5 by x+4 and combine like terms.

$$5x^{2}+25x+20=x+4+\left(x+1\right)\left(4x-2\right)$$

Use the distributive property to multiply x+1 by 4x-2 and combine like terms.

$$5x^{2}+25x+20=x+4+4x^{2}+2x-2$$

Combine x and 2x to get 3x.

$$5x^{2}+25x+20=3x+4+4x^{2}-2$$

Subtract 2 from 4 to get 2.

$$5x^{2}+25x+20=3x+2+4x^{2}$$

Subtract 3x from both sides.

$$5x^{2}+25x+20-3x=2+4x^{2}$$

Combine 25x and -3x to get 22x.

$$5x^{2}+22x+20=2+4x^{2}$$

Subtract 4x^{2} from both sides.

$$5x^{2}+22x+20-4x^{2}=2$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+22x+20=2$$

Subtract 20 from both sides.

$$x^{2}+22x=2-20$$

Subtract 20 from 2 to get -18.

$$x^{2}+22x=-18$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+22x+11^{2}=-18+11^{2}$$

Square 11.

$$x^{2}+22x+121=-18+121$$

Add -18 to 121.

$$x^{2}+22x+121=103$$

Factor x^{2}+22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11\right)^{2}=103$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{103}$$

Simplify.

$$x+11=\sqrt{103} x+11=-\sqrt{103}$$

Subtract 11 from both sides of the equation.

$$x=\sqrt{103}-11 x=-\sqrt{103}-11$$

Solve for x

$$x=\sqrt{103}-11\approx -0.851108435$$

$$x=-\sqrt{103}-11\approx -21.148891565$$

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+1,x+4.

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

Use the distributive property to multiply 5 by x+1.

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

Use the distributive property to multiply 5x+5 by x+4 and combine like terms.

$$5x^{2}+25x+20=x+4+\left(x+1\right)\left(4x-2\right)$$

Use the distributive property to multiply x+1 by 4x-2 and combine like terms.

$$5x^{2}+25x+20=x+4+4x^{2}+2x-2$$

Combine x and 2x to get 3x.

$$5x^{2}+25x+20=3x+4+4x^{2}-2$$

Subtract 2 from 4 to get 2.

$$5x^{2}+25x+20=3x+2+4x^{2}$$

Subtract 3x from both sides.

$$5x^{2}+25x+20-3x=2+4x^{2}$$

Combine 25x and -3x to get 22x.

$$5x^{2}+22x+20=2+4x^{2}$$

Subtract 2 from both sides.

$$5x^{2}+22x+20-2=4x^{2}$$

Subtract 2 from 20 to get 18.

$$5x^{2}+22x+18=4x^{2}$$

Subtract 4x^{2} from both sides.

$$5x^{2}+22x+18-4x^{2}=0$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+22x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 18}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-22±\sqrt{484-72}}{2}$$

Add 484 to -72.

$$x=\frac{-22±\sqrt{412}}{2}$$

Take the square root of 412.

$$x=\frac{-22±2\sqrt{103}}{2}$$

Now solve the equation x=\frac{-22±2\sqrt{103}}{2} when ± is plus. Add -22 to 2\sqrt{103}.

$$x=\frac{2\sqrt{103}-22}{2}$$

Divide -22+2\sqrt{103} by 2.

$$x=\sqrt{103}-11$$

Now solve the equation x=\frac{-22±2\sqrt{103}}{2} when ± is minus. Subtract 2\sqrt{103} from -22.

$$x=\frac{-2\sqrt{103}-22}{2}$$

Divide -22-2\sqrt{103} by 2.

$$x=-\sqrt{103}-11$$

The equation is now solved.

$$x=\sqrt{103}-11 x=-\sqrt{103}-11$$

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+1,x+4.

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

Use the distributive property to multiply 5 by x+1.

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

Use the distributive property to multiply 5x+5 by x+4 and combine like terms.

$$5x^{2}+25x+20=x+4+\left(x+1\right)\left(4x-2\right)$$

Use the distributive property to multiply x+1 by 4x-2 and combine like terms.

$$5x^{2}+25x+20=x+4+4x^{2}+2x-2$$

Combine x and 2x to get 3x.

$$5x^{2}+25x+20=3x+4+4x^{2}-2$$

Subtract 2 from 4 to get 2.

$$5x^{2}+25x+20=3x+2+4x^{2}$$

Subtract 3x from both sides.

$$5x^{2}+25x+20-3x=2+4x^{2}$$

Combine 25x and -3x to get 22x.

$$5x^{2}+22x+20=2+4x^{2}$$

Subtract 4x^{2} from both sides.

$$5x^{2}+22x+20-4x^{2}=2$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+22x+20=2$$

Subtract 20 from both sides.

$$x^{2}+22x=2-20$$

Subtract 20 from 2 to get -18.

$$x^{2}+22x=-18$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+22x+11^{2}=-18+11^{2}$$

Square 11.

$$x^{2}+22x+121=-18+121$$

Add -18 to 121.

$$x^{2}+22x+121=103$$

Factor x^{2}+22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11\right)^{2}=103$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{103}$$

Simplify.

$$x+11=\sqrt{103} x+11=-\sqrt{103}$$

Subtract 11 from both sides of the equation.

$$x=\sqrt{103}-11 x=-\sqrt{103}-11$$