Solve 20/x-5=60/x+5+1

$$\frac{ 20 }{ x-5 } = \frac{ 60 }{ x+5 } +1$$

Solve for x

$$x=5\sqrt{33}-20\approx 8.722813233$$

$$x=-5\sqrt{33}-20\approx -48.722813233$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x-5,x+5.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 20.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by 60.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$20x+100=60x-300+x^{2}-25$$

Subtract 25 from -300 to get -325.

$$20x+100=60x-325+x^{2}$$

Subtract 60x from both sides.

$$20x+100-60x=-325+x^{2}$$

Combine 20x and -60x to get -40x.

$$-40x+100=-325+x^{2}$$

Subtract -325 from both sides.

$$-40x+100-\left(-325\right)=x^{2}$$

The opposite of -325 is 325.

$$-40x+100+325=x^{2}$$

Subtract x^{2} from both sides.

$$-40x+100+325-x^{2}=0$$

Add 100 and 325 to get 425.

$$-40x+425-x^{2}=0$$

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.

$$-x^{2}-40x+425=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -40 for b, and 425 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-40\right)±\sqrt{1600+4\times 425}}{2\left(-1\right)}$$

Multiply 4 times 425.

$$x=\frac{-\left(-40\right)±\sqrt{1600+1700}}{2\left(-1\right)}$$

Add 1600 to 1700.

$$x=\frac{-\left(-40\right)±\sqrt{3300}}{2\left(-1\right)}$$

Take the square root of 3300.

$$x=\frac{-\left(-40\right)±10\sqrt{33}}{2\left(-1\right)}$$

The opposite of -40 is 40.

$$x=\frac{40±10\sqrt{33}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{40±10\sqrt{33}}{-2}$$

Now solve the equation x=\frac{40±10\sqrt{33}}{-2} when ± is plus. Add 40 to 10\sqrt{33}.

$$x=\frac{10\sqrt{33}+40}{-2}$$

Divide 40+10\sqrt{33} by -2.

$$x=-5\sqrt{33}-20$$

Now solve the equation x=\frac{40±10\sqrt{33}}{-2} when ± is minus. Subtract 10\sqrt{33} from 40.

$$x=\frac{40-10\sqrt{33}}{-2}$$

Divide 40-10\sqrt{33} by -2.

$$x=5\sqrt{33}-20$$

The equation is now solved.

$$x=-5\sqrt{33}-20 x=5\sqrt{33}-20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x-5,x+5.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 20.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by 60.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$20x+100=60x-300+x^{2}-25$$

Subtract 25 from -300 to get -325.

$$20x+100=60x-325+x^{2}$$

Subtract 60x from both sides.

$$20x+100-60x=-325+x^{2}$$

Combine 20x and -60x to get -40x.

$$-40x+100=-325+x^{2}$$

Subtract x^{2} from both sides.

$$-40x+100-x^{2}=-325$$

Subtract 100 from both sides.

$$-40x-x^{2}=-325-100$$

Subtract 100 from -325 to get -425.

$$-40x-x^{2}=-425$$

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.

$$-x^{2}-40x=-425$$

Divide both sides by -1.

$$\frac{-x^{2}-40x}{-1}=-\frac{425}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{40}{-1}\right)x=-\frac{425}{-1}$$

Divide -40 by -1.

$$x^{2}+40x=-\frac{425}{-1}$$

Divide -425 by -1.

$$x^{2}+40x=425$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+40x+20^{2}=425+20^{2}$$

Square 20.

$$x^{2}+40x+400=425+400$$

Add 425 to 400.

$$x^{2}+40x+400=825$$

Factor x^{2}+40x+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}.

$$\left(x+20\right)^{2}=825$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{825}$$

Simplify.

$$x+20=5\sqrt{33} x+20=-5\sqrt{33}$$

Subtract 20 from both sides of the equation.

$$x=5\sqrt{33}-20 x=-5\sqrt{33}-20$$