Solve 15/x+10/x-10=1

$$\frac{ 15 }{ x } + \frac{ 10 }{ x-10 } = 1$$

Solve for x

$$x=5$$

$$x=30$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,10 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,x-10.

$$\left(x-10\right)\times 15+x\times 10=x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 15.

$$15x-150+x\times 10=x\left(x-10\right)$$

Combine 15x and x\times 10 to get 25x.

$$25x-150=x\left(x-10\right)$$

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

$$25x-150=x^{2}-10x$$

Subtract x^{2} from both sides.

$$25x-150-x^{2}=-10x$$

Add 10x to both sides.

$$25x-150-x^{2}+10x=0$$

Combine 25x and 10x to get 35x.

$$35x-150-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+35x-150=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-150. To find a and b, set up a system to be solved.

$$a+b=35 ab=-\left(-150\right)=150$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 150.

$$1,150 2,75 3,50 5,30 6,25 10,15$$

Calculate the sum for each pair.

$$1+150=151 2+75=77 3+50=53 5+30=35 6+25=31 10+15=25$$

The solution is the pair that gives sum 35.

$$a=30 b=5$$

Rewrite -x^{2}+35x-150 as \left(-x^{2}+30x\right)+\left(5x-150\right).

$$\left(-x^{2}+30x\right)+\left(5x-150\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-30\right)+5\left(x-30\right)$$

Factor out common term x-30 by using distributive property.

$$\left(x-30\right)\left(-x+5\right)$$

To find equation solutions, solve x-30=0 and -x+5=0.

$$x=30 x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 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,x-10.

$$\left(x-10\right)\times 15+x\times 10=x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 15.

$$15x-150+x\times 10=x\left(x-10\right)$$

Combine 15x and x\times 10 to get 25x.

$$25x-150=x\left(x-10\right)$$

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

$$25x-150=x^{2}-10x$$

Subtract x^{2} from both sides.

$$25x-150-x^{2}=-10x$$

Add 10x to both sides.

$$25x-150-x^{2}+10x=0$$

Combine 25x and 10x to get 35x.

$$35x-150-x^{2}=0$$

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.

$$-x^{2}+35x-150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 35 for b, and -150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\left(-1\right)\left(-150\right)}}{2\left(-1\right)}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\left(-1\right)\left(-150\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-35±\sqrt{1225+4\left(-150\right)}}{2\left(-1\right)}$$

Multiply 4 times -150.

$$x=\frac{-35±\sqrt{1225-600}}{2\left(-1\right)}$$

Add 1225 to -600.

$$x=\frac{-35±\sqrt{625}}{2\left(-1\right)}$$

Take the square root of 625.

$$x=\frac{-35±25}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-35±25}{-2}$$

Now solve the equation x=\frac{-35±25}{-2} when ± is plus. Add -35 to 25.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-35±25}{-2} when ± is minus. Subtract 25 from -35.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

The equation is now solved.

$$x=5 x=30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 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,x-10.

$$\left(x-10\right)\times 15+x\times 10=x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 15.

$$15x-150+x\times 10=x\left(x-10\right)$$

Combine 15x and x\times 10 to get 25x.

$$25x-150=x\left(x-10\right)$$

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

$$25x-150=x^{2}-10x$$

Subtract x^{2} from both sides.

$$25x-150-x^{2}=-10x$$

Add 10x to both sides.

$$25x-150-x^{2}+10x=0$$

Combine 25x and 10x to get 35x.

$$35x-150-x^{2}=0$$

Add 150 to both sides. Anything plus zero gives itself.

$$35x-x^{2}=150$$

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.

$$-x^{2}+35x=150$$

Divide both sides by -1.

$$\frac{-x^{2}+35x}{-1}=\frac{150}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{35}{-1}x=\frac{150}{-1}$$

Divide 35 by -1.

$$x^{2}-35x=\frac{150}{-1}$$

Divide 150 by -1.

$$x^{2}-35x=-150$$

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

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-150+\left(-\frac{35}{2}\right)^{2}$$

Square -\frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-35x+\frac{1225}{4}=-150+\frac{1225}{4}$$

Add -150 to \frac{1225}{4}.

$$x^{2}-35x+\frac{1225}{4}=\frac{625}{4}$$

Factor x^{2}-35x+\frac{1225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$x-\frac{35}{2}=\frac{25}{2} x-\frac{35}{2}=-\frac{25}{2}$$

Add \frac{35}{2} to both sides of the equation.

$$x=30 x=5$$