Solve 154/1+x-90/1-x=40

$$\frac{ 154 }{ 1+x } - \frac{ 90 }{ 1-x } = 40$$

Solve for x

$$x=\frac{1}{10}=0.1$$

$$x=6$$

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 1+x,1-x.

$$\left(x-1\right)\times 154-\left(-\left(1+x\right)\times 90\right)=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 154.

$$154x-154-\left(-\left(1+x\right)\times 90\right)=40\left(x-1\right)\left(x+1\right)$$

Multiply -1 and 90 to get -90.

$$154x-154-\left(-90\left(1+x\right)\right)=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -90 by 1+x.

$$154x-154-\left(-90-90x\right)=40\left(x-1\right)\left(x+1\right)$$

To find the opposite of -90-90x, find the opposite of each term.

$$154x-154+90+90x=40\left(x-1\right)\left(x+1\right)$$

Add -154 and 90 to get -64.

$$154x-64+90x=40\left(x-1\right)\left(x+1\right)$$

Combine 154x and 90x to get 244x.

$$244x-64=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 40 by x-1.

$$244x-64=\left(40x-40\right)\left(x+1\right)$$

Use the distributive property to multiply 40x-40 by x+1 and combine like terms.

$$244x-64=40x^{2}-40$$

Subtract 40x^{2} from both sides.

$$244x-64-40x^{2}=-40$$

Add 40 to both sides.

$$244x-64-40x^{2}+40=0$$

Add -64 and 40 to get -24.

$$244x-24-40x^{2}=0$$

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.

$$-40x^{2}+244x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -40 for a, 244 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-244±\sqrt{244^{2}-4\left(-40\right)\left(-24\right)}}{2\left(-40\right)}$$

Square 244.

$$x=\frac{-244±\sqrt{59536-4\left(-40\right)\left(-24\right)}}{2\left(-40\right)}$$

Multiply -4 times -40.

$$x=\frac{-244±\sqrt{59536+160\left(-24\right)}}{2\left(-40\right)}$$

Multiply 160 times -24.

$$x=\frac{-244±\sqrt{59536-3840}}{2\left(-40\right)}$$

Add 59536 to -3840.

$$x=\frac{-244±\sqrt{55696}}{2\left(-40\right)}$$

Take the square root of 55696.

$$x=\frac{-244±236}{2\left(-40\right)}$$

Multiply 2 times -40.

$$x=\frac{-244±236}{-80}$$

Now solve the equation x=\frac{-244±236}{-80} when ± is plus. Add -244 to 236.

$$x=-\frac{8}{-80}$$

Reduce the fraction \frac{-8}{-80} to lowest terms by extracting and canceling out 8.

$$x=\frac{1}{10}$$

Now solve the equation x=\frac{-244±236}{-80} when ± is minus. Subtract 236 from -244.

$$x=-\frac{480}{-80}$$

Divide -480 by -80.

$$x=6$$

The equation is now solved.

$$x=\frac{1}{10} x=6$$

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 1+x,1-x.

$$\left(x-1\right)\times 154-\left(-\left(1+x\right)\times 90\right)=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-1 by 154.

$$154x-154-\left(-\left(1+x\right)\times 90\right)=40\left(x-1\right)\left(x+1\right)$$

Multiply -1 and 90 to get -90.

$$154x-154-\left(-90\left(1+x\right)\right)=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -90 by 1+x.

$$154x-154-\left(-90-90x\right)=40\left(x-1\right)\left(x+1\right)$$

To find the opposite of -90-90x, find the opposite of each term.

$$154x-154+90+90x=40\left(x-1\right)\left(x+1\right)$$

Add -154 and 90 to get -64.

$$154x-64+90x=40\left(x-1\right)\left(x+1\right)$$

Combine 154x and 90x to get 244x.

$$244x-64=40\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 40 by x-1.

$$244x-64=\left(40x-40\right)\left(x+1\right)$$

Use the distributive property to multiply 40x-40 by x+1 and combine like terms.

$$244x-64=40x^{2}-40$$

Subtract 40x^{2} from both sides.

$$244x-64-40x^{2}=-40$$

Add 64 to both sides.

$$244x-40x^{2}=-40+64$$

Add -40 and 64 to get 24.

$$244x-40x^{2}=24$$

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.

$$-40x^{2}+244x=24$$

Divide both sides by -40.

$$\frac{-40x^{2}+244x}{-40}=\frac{24}{-40}$$

Dividing by -40 undoes the multiplication by -40.

$$x^{2}+\frac{244}{-40}x=\frac{24}{-40}$$

Reduce the fraction \frac{244}{-40} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{61}{10}x=\frac{24}{-40}$$

Reduce the fraction \frac{24}{-40} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{61}{10}x=-\frac{3}{5}$$

Divide -\frac{61}{10}, the coefficient of the x term, by 2 to get -\frac{61}{20}. Then add the square of -\frac{61}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{61}{10}x+\left(-\frac{61}{20}\right)^{2}=-\frac{3}{5}+\left(-\frac{61}{20}\right)^{2}$$

Square -\frac{61}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{61}{10}x+\frac{3721}{400}=-\frac{3}{5}+\frac{3721}{400}$$

Add -\frac{3}{5} to \frac{3721}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{61}{10}x+\frac{3721}{400}=\frac{3481}{400}$$

Factor x^{2}-\frac{61}{10}x+\frac{3721}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{61}{20}\right)^{2}=\frac{3481}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{61}{20}\right)^{2}}=\sqrt{\frac{3481}{400}}$$

Simplify.

$$x-\frac{61}{20}=\frac{59}{20} x-\frac{61}{20}=-\frac{59}{20}$$

Add \frac{61}{20} to both sides of the equation.

$$x=6 x=\frac{1}{10}$$