Solve x+4/x+1-10/x^2-1=8

$$\frac{ x+4 }{ x+1 } - \frac{ 10 }{ { x }^{ 2 } -1 } =8$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{159}i+3}{14}\approx 0.214285714-0.900680015i$$

$$x=\frac{3+\sqrt{159}i}{14}\approx 0.214285714+0.900680015i$$

Steps Using the Quadratic Formula

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

$$\left(x-1\right)\left(x+4\right)-10=8\left(x-1\right)\left(x+1\right)$$

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

$$x^{2}+3x-4-10=8\left(x-1\right)\left(x+1\right)$$

Subtract 10 from -4 to get -14.

$$x^{2}+3x-14=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 8 by x-1.

$$x^{2}+3x-14=\left(8x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-8 by x+1 and combine like terms.

$$x^{2}+3x-14=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$x^{2}+3x-14-8x^{2}=-8$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}+3x-14=-8$$

Add 8 to both sides.

$$-7x^{2}+3x-14+8=0$$

Add -14 and 8 to get -6.

$$-7x^{2}+3x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 3 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-7\right)\left(-6\right)}}{2\left(-7\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-7\right)\left(-6\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-3±\sqrt{9+28\left(-6\right)}}{2\left(-7\right)}$$

Multiply 28 times -6.

$$x=\frac{-3±\sqrt{9-168}}{2\left(-7\right)}$$

Add 9 to -168.

$$x=\frac{-3±\sqrt{-159}}{2\left(-7\right)}$$

Take the square root of -159.

$$x=\frac{-3±\sqrt{159}i}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-3±\sqrt{159}i}{-14}$$

Now solve the equation x=\frac{-3±\sqrt{159}i}{-14} when ± is plus. Add -3 to i\sqrt{159}.

$$x=\frac{-3+\sqrt{159}i}{-14}$$

Divide -3+i\sqrt{159} by -14.

$$x=\frac{-\sqrt{159}i+3}{14}$$

Now solve the equation x=\frac{-3±\sqrt{159}i}{-14} when ± is minus. Subtract i\sqrt{159} from -3.

$$x=\frac{-\sqrt{159}i-3}{-14}$$

Divide -3-i\sqrt{159} by -14.

$$x=\frac{3+\sqrt{159}i}{14}$$

The equation is now solved.

$$x=\frac{-\sqrt{159}i+3}{14} x=\frac{3+\sqrt{159}i}{14}$$

Steps for Completing the Square

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

$$\left(x-1\right)\left(x+4\right)-10=8\left(x-1\right)\left(x+1\right)$$

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

$$x^{2}+3x-4-10=8\left(x-1\right)\left(x+1\right)$$

Subtract 10 from -4 to get -14.

$$x^{2}+3x-14=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 8 by x-1.

$$x^{2}+3x-14=\left(8x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-8 by x+1 and combine like terms.

$$x^{2}+3x-14=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$x^{2}+3x-14-8x^{2}=-8$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}+3x-14=-8$$

Add 14 to both sides.

$$-7x^{2}+3x=-8+14$$

Add -8 and 14 to get 6.

$$-7x^{2}+3x=6$$

Divide both sides by -7.

$$\frac{-7x^{2}+3x}{-7}=\frac{6}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{3}{-7}x=\frac{6}{-7}$$

Divide 3 by -7.

$$x^{2}-\frac{3}{7}x=\frac{6}{-7}$$

Divide 6 by -7.

$$x^{2}-\frac{3}{7}x=-\frac{6}{7}$$

Divide -\frac{3}{7}, the coefficient of the x term, by 2 to get -\frac{3}{14}. Then add the square of -\frac{3}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=-\frac{6}{7}+\left(-\frac{3}{14}\right)^{2}$$

Square -\frac{3}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=-\frac{6}{7}+\frac{9}{196}$$

Add -\frac{6}{7} to \frac{9}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=-\frac{159}{196}$$

Factor x^{2}-\frac{3}{7}x+\frac{9}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{14}\right)^{2}=-\frac{159}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{-\frac{159}{196}}$$

Simplify.

$$x-\frac{3}{14}=\frac{\sqrt{159}i}{14} x-\frac{3}{14}=-\frac{\sqrt{159}i}{14}$$

Add \frac{3}{14} to both sides of the equation.

$$x=\frac{3+\sqrt{159}i}{14} x=\frac{-\sqrt{159}i+3}{14}$$