Solve 4/x-1+2/x+1=35

$$\frac { 4 } { x - 1 } + \frac { 2 } { x + 1 } = 35$$

Solve for x

$$x = \frac{2 \sqrt{326} + 3}{35} \approx 1.117455433$$

$$x=\frac{3-2\sqrt{326}}{35}\approx -0.946026862$$

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+1.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 4.

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 2.

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

Combine 4x and 2x to get 6x.

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 4 to get 2.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 35 by x-1.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

Use the distributive property to multiply 35x-35 by x+1 and combine like terms.

$$6x+2=35x^{2}-35$$

Subtract 35x^{2} from both sides.

$$6x+2-35x^{2}=-35$$

Add 35 to both sides.

$$6x+2-35x^{2}+35=0$$

Add 2 and 35 to get 37.

$$6x+37-35x^{2}=0$$

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.

$$-35x^{2}+6x+37=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -35 for a, 6 for b, and 37 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

Multiply -4 times -35.

$$x=\frac{-6±\sqrt{36+140\times 37}}{2\left(-35\right)}$$

Multiply 140 times 37.

$$x=\frac{-6±\sqrt{36+5180}}{2\left(-35\right)}$$

Add 36 to 5180.

$$x=\frac{-6±\sqrt{5216}}{2\left(-35\right)}$$

Take the square root of 5216.

$$x=\frac{-6±4\sqrt{326}}{2\left(-35\right)}$$

Multiply 2 times -35.

$$x=\frac{-6±4\sqrt{326}}{-70}$$

Now solve the equation x=\frac{-6±4\sqrt{326}}{-70} when ± is plus. Add -6 to 4\sqrt{326}.

$$x=\frac{4\sqrt{326}-6}{-70}$$

Divide -6+4\sqrt{326} by -70.

$$x=\frac{3-2\sqrt{326}}{35}$$

Now solve the equation x=\frac{-6±4\sqrt{326}}{-70} when ± is minus. Subtract 4\sqrt{326} from -6.

$$x=\frac{-4\sqrt{326}-6}{-70}$$

Divide -6-4\sqrt{326} by -70.

$$x=\frac{2\sqrt{326}+3}{35}$$

The equation is now solved.

$$x=\frac{3-2\sqrt{326}}{35} x=\frac{2\sqrt{326}+3}{35}$$

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+1.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x+1 by 4.

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 2.

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

Combine 4x and 2x to get 6x.

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

Subtract 2 from 4 to get 2.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 35 by x-1.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

Use the distributive property to multiply 35x-35 by x+1 and combine like terms.

$$6x+2=35x^{2}-35$$

Subtract 35x^{2} from both sides.

$$6x+2-35x^{2}=-35$$

Subtract 2 from both sides.

$$6x-35x^{2}=-35-2$$

Subtract 2 from -35 to get -37.

$$6x-35x^{2}=-37$$

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.

$$-35x^{2}+6x=-37$$

Divide both sides by -35.

$$\frac{-35x^{2}+6x}{-35}=-\frac{37}{-35}$$

Dividing by -35 undoes the multiplication by -35.

$$x^{2}+\frac{6}{-35}x=-\frac{37}{-35}$$

Divide 6 by -35.

$$x^{2}-\frac{6}{35}x=-\frac{37}{-35}$$

Divide -37 by -35.

$$x^{2}-\frac{6}{35}x=\frac{37}{35}$$

Divide -\frac{6}{35}, the coefficient of the x term, by 2 to get -\frac{3}{35}. Then add the square of -\frac{3}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{35}x+\left(-\frac{3}{35}\right)^{2}=\frac{37}{35}+\left(-\frac{3}{35}\right)^{2}$$

Square -\frac{3}{35} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{37}{35}+\frac{9}{1225}$$

Add \frac{37}{35} to \frac{9}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{1304}{1225}$$

Factor x^{2}-\frac{6}{35}x+\frac{9}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{35}\right)^{2}=\frac{1304}{1225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{35}\right)^{2}}=\sqrt{\frac{1304}{1225}}$$

Simplify.

$$x-\frac{3}{35}=\frac{2\sqrt{326}}{35} x-\frac{3}{35}=-\frac{2\sqrt{326}}{35}$$

Add \frac{3}{35} to both sides of the equation.

$$x=\frac{2\sqrt{326}+3}{35} x=\frac{3-2\sqrt{326}}{35}$$