Solve 90/x+90-40/x+40=0.2

$$\frac{ 90 }{ x+90 } - \frac{ 40 }{ x+40 } =0.2$$

Solve for x

$$x=60$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -90,-40 since division by zero is not defined. Multiply both sides of the equation by \left(x+40\right)\left(x+90\right), the least common multiple of x+90,x+40.

$$\left(x+40\right)\times 90-\left(x+90\right)\times 40=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply x+40 by 90.

$$90x+3600-\left(x+90\right)\times 40=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply x+90 by 40.

$$90x+3600-\left(40x+3600\right)=0.2\left(x+40\right)\left(x+90\right)$$

To find the opposite of 40x+3600, find the opposite of each term.

$$90x+3600-40x-3600=0.2\left(x+40\right)\left(x+90\right)$$

Combine 90x and -40x to get 50x.

$$50x+3600-3600=0.2\left(x+40\right)\left(x+90\right)$$

Subtract 3600 from 3600 to get 0.

$$50x=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply 0.2 by x+40.

$$50x=\left(0.2x+8\right)\left(x+90\right)$$

Use the distributive property to multiply 0.2x+8 by x+90 and combine like terms.

$$50x=0.2x^{2}+26x+720$$

Subtract 0.2x^{2} from both sides.

$$50x-0.2x^{2}=26x+720$$

Subtract 26x from both sides.

$$50x-0.2x^{2}-26x=720$$

Combine 50x and -26x to get 24x.

$$24x-0.2x^{2}=720$$

Subtract 720 from both sides.

$$24x-0.2x^{2}-720=0$$

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.

$$-0.2x^{2}+24x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.2 for a, 24 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-0.2\right)\left(-720\right)}}{2\left(-0.2\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-0.2\right)\left(-720\right)}}{2\left(-0.2\right)}$$

Multiply -4 times -0.2.

$$x=\frac{-24±\sqrt{576+0.8\left(-720\right)}}{2\left(-0.2\right)}$$

Multiply 0.8 times -720.

$$x=\frac{-24±\sqrt{576-576}}{2\left(-0.2\right)}$$

Add 576 to -576.

$$x=\frac{-24±\sqrt{0}}{2\left(-0.2\right)}$$

Take the square root of 0.

$$x=-\frac{24}{2\left(-0.2\right)}$$

Multiply 2 times -0.2.

$$x=-\frac{24}{-0.4}$$

Divide -24 by -0.4 by multiplying -24 by the reciprocal of -0.4.

$$x=60$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -90,-40 since division by zero is not defined. Multiply both sides of the equation by \left(x+40\right)\left(x+90\right), the least common multiple of x+90,x+40.

$$\left(x+40\right)\times 90-\left(x+90\right)\times 40=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply x+40 by 90.

$$90x+3600-\left(x+90\right)\times 40=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply x+90 by 40.

$$90x+3600-\left(40x+3600\right)=0.2\left(x+40\right)\left(x+90\right)$$

To find the opposite of 40x+3600, find the opposite of each term.

$$90x+3600-40x-3600=0.2\left(x+40\right)\left(x+90\right)$$

Combine 90x and -40x to get 50x.

$$50x+3600-3600=0.2\left(x+40\right)\left(x+90\right)$$

Subtract 3600 from 3600 to get 0.

$$50x=0.2\left(x+40\right)\left(x+90\right)$$

Use the distributive property to multiply 0.2 by x+40.

$$50x=\left(0.2x+8\right)\left(x+90\right)$$

Use the distributive property to multiply 0.2x+8 by x+90 and combine like terms.

$$50x=0.2x^{2}+26x+720$$

Subtract 0.2x^{2} from both sides.

$$50x-0.2x^{2}=26x+720$$

Subtract 26x from both sides.

$$50x-0.2x^{2}-26x=720$$

Combine 50x and -26x to get 24x.

$$24x-0.2x^{2}=720$$

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.

$$-0.2x^{2}+24x=720$$

Multiply both sides by -5.

$$\frac{-0.2x^{2}+24x}{-0.2}=\frac{720}{-0.2}$$

Dividing by -0.2 undoes the multiplication by -0.2.

$$x^{2}+\frac{24}{-0.2}x=\frac{720}{-0.2}$$

Divide 24 by -0.2 by multiplying 24 by the reciprocal of -0.2.

$$x^{2}-120x=\frac{720}{-0.2}$$

Divide 720 by -0.2 by multiplying 720 by the reciprocal of -0.2.

$$x^{2}-120x=-3600$$

Divide -120, the coefficient of the x term, by 2 to get -60. Then add the square of -60 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-120x+\left(-60\right)^{2}=-3600+\left(-60\right)^{2}$$

Square -60.

$$x^{2}-120x+3600=-3600+3600$$

Add -3600 to 3600.

$$x^{2}-120x+3600=0$$

Factor x^{2}-120x+3600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-60\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-60\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-60=0 x-60=0$$

Add 60 to both sides of the equation.

$$x=60 x=60$$

The equation is now solved. Solutions are the same.

$$x=60$$