Solve 8x+4/x^3+1+4/x+1-5x-1/x^2-x+1=0

$$\frac { 8 x + 4 } { x ^ { 3 } + 1 } + \frac { 4 } { x + 1 } - \frac { 5 x - 1 } { x ^ { 2 } - x + 1 } = 0$$

Solve for x

$$x=3$$

$$x=-3$$

Steps by Finding Square Root

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

$$8x+4+\left(x^{2}-x+1\right)\times 4-\left(x+1\right)\left(5x-1\right)=0$$

Use the distributive property to multiply x^{2}-x+1 by 4.

$$8x+4+4x^{2}-4x+4-\left(x+1\right)\left(5x-1\right)=0$$

Combine 8x and -4x to get 4x.

$$4x+4+4x^{2}+4-\left(x+1\right)\left(5x-1\right)=0$$

Add 4 and 4 to get 8.

$$4x+8+4x^{2}-\left(x+1\right)\left(5x-1\right)=0$$

Use the distributive property to multiply x+1 by 5x-1 and combine like terms.

$$4x+8+4x^{2}-\left(5x^{2}+4x-1\right)=0$$

To find the opposite of 5x^{2}+4x-1, find the opposite of each term.

$$4x+8+4x^{2}-5x^{2}-4x+1=0$$

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

$$4x+8-x^{2}-4x+1=0$$

Combine 4x and -4x to get 0.

$$8-x^{2}+1=0$$

Add 8 and 1 to get 9.

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

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -1.

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

Fraction \frac{-9}{-1} can be simplified to 9 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=9$$

Take the square root of both sides of the equation.

$$x=3 x=-3$$

Steps Using the Quadratic Formula

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

$$8x+4+\left(x^{2}-x+1\right)\times 4-\left(x+1\right)\left(5x-1\right)=0$$

Use the distributive property to multiply x^{2}-x+1 by 4.

$$8x+4+4x^{2}-4x+4-\left(x+1\right)\left(5x-1\right)=0$$

Combine 8x and -4x to get 4x.

$$4x+4+4x^{2}+4-\left(x+1\right)\left(5x-1\right)=0$$

Add 4 and 4 to get 8.

$$4x+8+4x^{2}-\left(x+1\right)\left(5x-1\right)=0$$

Use the distributive property to multiply x+1 by 5x-1 and combine like terms.

$$4x+8+4x^{2}-\left(5x^{2}+4x-1\right)=0$$

To find the opposite of 5x^{2}+4x-1, find the opposite of each term.

$$4x+8+4x^{2}-5x^{2}-4x+1=0$$

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

$$4x+8-x^{2}-4x+1=0$$

Combine 4x and -4x to get 0.

$$8-x^{2}+1=0$$

Add 8 and 1 to get 9.

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 9}}{2\left(-1\right)}$$

Multiply 4 times 9.

$$x=\frac{0±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$x=\frac{0±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±6}{-2}$$

Now solve the equation x=\frac{0±6}{-2} when ± is plus. Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{0±6}{-2} when ± is minus. Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-3 x=3$$