Solve 200/(x+10)=200/x-1

$$\frac{ 200 }{ (x+10) } = \frac{ 200 }{ x } -1$$

Solve for x

$$x=-50$$

$$x=40$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x\times 200=\left(x+10\right)\times 200+x\left(x+10\right)\left(-1\right)$$

Use the distributive property to multiply x+10 by 200.

$$x\times 200=200x+2000+x\left(x+10\right)\left(-1\right)$$

Use the distributive property to multiply x by x+10.

$$x\times 200=200x+2000+\left(x^{2}+10x\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}+10x by -1.

$$x\times 200=200x+2000-x^{2}-10x$$

Combine 200x and -10x to get 190x.

$$x\times 200=190x+2000-x^{2}$$

Subtract 190x from both sides.

$$x\times 200-190x=2000-x^{2}$$

Combine x\times 200 and -190x to get 10x.

$$10x=2000-x^{2}$$

Subtract 2000 from both sides.

$$10x-2000=-x^{2}$$

Add x^{2} to both sides.

$$10x-2000+x^{2}=0$$

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.

$$x^{2}+10x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2000\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2000\right)}}{2}$$

Multiply -4 times -2000.

$$x=\frac{-10±\sqrt{100+8000}}{2}$$

Add 100 to 8000.

$$x=\frac{-10±\sqrt{8100}}{2}$$

Take the square root of 8100.

$$x=\frac{-10±90}{2}$$

Now solve the equation x=\frac{-10±90}{2} when ± is plus. Add -10 to 90.

$$x=\frac{80}{2}$$

Divide 80 by 2.

$$x=40$$

Now solve the equation x=\frac{-10±90}{2} when ± is minus. Subtract 90 from -10.

$$x=-\frac{100}{2}$$

Divide -100 by 2.

$$x=-50$$

The equation is now solved.

$$x=40 x=-50$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x+10,x.

$$x\times 200=\left(x+10\right)\times 200+x\left(x+10\right)\left(-1\right)$$

Use the distributive property to multiply x+10 by 200.

$$x\times 200=200x+2000+x\left(x+10\right)\left(-1\right)$$

Use the distributive property to multiply x by x+10.

$$x\times 200=200x+2000+\left(x^{2}+10x\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}+10x by -1.

$$x\times 200=200x+2000-x^{2}-10x$$

Combine 200x and -10x to get 190x.

$$x\times 200=190x+2000-x^{2}$$

Subtract 190x from both sides.

$$x\times 200-190x=2000-x^{2}$$

Combine x\times 200 and -190x to get 10x.

$$10x=2000-x^{2}$$

Add x^{2} to both sides.

$$10x+x^{2}=2000$$

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.

$$x^{2}+10x=2000$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=2000+5^{2}$$

Square 5.

$$x^{2}+10x+25=2000+25$$

Add 2000 to 25.

$$x^{2}+10x+25=2025$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=2025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{2025}$$

Simplify.

$$x+5=45 x+5=-45$$

Subtract 5 from both sides of the equation.

$$x=40 x=-50$$