Solve 1/x+1+4/(1-x)(x+1)+3/2(1-x)=4

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{159}i}{16}\approx -0.0625+0.788095013i$$

$$x=\frac{-\sqrt{159}i-1}{16}\approx -0.0625-0.788095013i$$

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 2\left(x-1\right)\left(x+1\right), the least common multiple of x+1,\left(1-x\right)\left(x+1\right),2\left(1-x\right).

$$2x-2-2\times 4-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Multiply -2 and 4 to get -8.

$$2x-2-8-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Subtract 8 from -2 to get -10.

$$2x-10-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Multiply -1 and 3 to get -3.

$$2x-10-3\left(1+x\right)=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3 by 1+x.

$$2x-10-3-3x=8\left(x-1\right)\left(x+1\right)$$

Subtract 3 from -10 to get -13.

$$2x-13-3x=8\left(x-1\right)\left(x+1\right)$$

Combine 2x and -3x to get -x.

$$-x-13=8\left(x-1\right)\left(x+1\right)$$

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

$$-x-13=\left(8x-8\right)\left(x+1\right)$$

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

$$-x-13=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$-x-13-8x^{2}=-8$$

Add 8 to both sides.

$$-x-13-8x^{2}+8=0$$

Add -13 and 8 to get -5.

$$-x-5-8x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8x^{2}-x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -1 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-8\right)\left(-5\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-1\right)±\sqrt{1+32\left(-5\right)}}{2\left(-8\right)}$$

Multiply 32 times -5.

$$x=\frac{-\left(-1\right)±\sqrt{1-160}}{2\left(-8\right)}$$

Add 1 to -160.

$$x=\frac{-\left(-1\right)±\sqrt{-159}}{2\left(-8\right)}$$

Take the square root of -159.

$$x=\frac{-\left(-1\right)±\sqrt{159}i}{2\left(-8\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{159}i}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{1±\sqrt{159}i}{-16}$$

Now solve the equation x=\frac{1±\sqrt{159}i}{-16} when ± is plus. Add 1 to i\sqrt{159}.

$$x=\frac{1+\sqrt{159}i}{-16}$$

Divide 1+i\sqrt{159} by -16.

$$x=\frac{-\sqrt{159}i-1}{16}$$

Now solve the equation x=\frac{1±\sqrt{159}i}{-16} when ± is minus. Subtract i\sqrt{159} from 1.

$$x=\frac{-\sqrt{159}i+1}{-16}$$

Divide 1-i\sqrt{159} by -16.

$$x=\frac{-1+\sqrt{159}i}{16}$$

The equation is now solved.

$$x=\frac{-\sqrt{159}i-1}{16} x=\frac{-1+\sqrt{159}i}{16}$$

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 2\left(x-1\right)\left(x+1\right), the least common multiple of x+1,\left(1-x\right)\left(x+1\right),2\left(1-x\right).

$$2x-2-2\times 4-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Multiply -2 and 4 to get -8.

$$2x-2-8-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Subtract 8 from -2 to get -10.

$$2x-10-\left(1+x\right)\times 3=8\left(x-1\right)\left(x+1\right)$$

Multiply -1 and 3 to get -3.

$$2x-10-3\left(1+x\right)=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3 by 1+x.

$$2x-10-3-3x=8\left(x-1\right)\left(x+1\right)$$

Subtract 3 from -10 to get -13.

$$2x-13-3x=8\left(x-1\right)\left(x+1\right)$$

Combine 2x and -3x to get -x.

$$-x-13=8\left(x-1\right)\left(x+1\right)$$

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

$$-x-13=\left(8x-8\right)\left(x+1\right)$$

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

$$-x-13=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$-x-13-8x^{2}=-8$$

Add 13 to both sides.

$$-x-8x^{2}=-8+13$$

Add -8 and 13 to get 5.

$$-x-8x^{2}=5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}-x=5$$

Divide both sides by -8.

$$\frac{-8x^{2}-x}{-8}=\frac{5}{-8}$$

Dividing by -8 undoes the multiplication by -8.

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

Divide -1 by -8.

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

Divide 5 by -8.

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

Divide \frac{1}{8}, the coefficient of the x term, by 2 to get \frac{1}{16}. Then add the square of \frac{1}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{8}x+\left(\frac{1}{16}\right)^{2}=-\frac{5}{8}+\left(\frac{1}{16}\right)^{2}$$

Square \frac{1}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{8}x+\frac{1}{256}=-\frac{5}{8}+\frac{1}{256}$$

Add -\frac{5}{8} to \frac{1}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{8}x+\frac{1}{256}=-\frac{159}{256}$$

Factor x^{2}+\frac{1}{8}x+\frac{1}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{16}\right)^{2}=-\frac{159}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{16}\right)^{2}}=\sqrt{-\frac{159}{256}}$$

Simplify.

$$x+\frac{1}{16}=\frac{\sqrt{159}i}{16} x+\frac{1}{16}=-\frac{\sqrt{159}i}{16}$$

Subtract \frac{1}{16} from both sides of the equation.

$$x=\frac{-1+\sqrt{159}i}{16} x=\frac{-\sqrt{159}i-1}{16}$$