Solve 15/x+15/x-50=3

$$\frac{ 15 }{ x } + \frac{ 15 }{ x-50 } = 3$$

Solve for x

$$x=5\sqrt{26}+30\approx 55.495097568$$

$$x=30-5\sqrt{26}\approx 4.504902432$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,50 since division by zero is not defined. Multiply both sides of the equation by x\left(x-50\right), the least common multiple of x,x-50.

$$\left(x-50\right)\times 15+x\times 15=3x\left(x-50\right)$$

Use the distributive property to multiply x-50 by 15.

$$15x-750+x\times 15=3x\left(x-50\right)$$

Combine 15x and x\times 15 to get 30x.

$$30x-750=3x\left(x-50\right)$$

Use the distributive property to multiply 3x by x-50.

$$30x-750=3x^{2}-150x$$

Subtract 3x^{2} from both sides.

$$30x-750-3x^{2}=-150x$$

Add 150x to both sides.

$$30x-750-3x^{2}+150x=0$$

Combine 30x and 150x to get 180x.

$$180x-750-3x^{2}=0$$

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.

$$-3x^{2}+180x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 180 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-180±\sqrt{180^{2}-4\left(-3\right)\left(-750\right)}}{2\left(-3\right)}$$

Square 180.

$$x=\frac{-180±\sqrt{32400-4\left(-3\right)\left(-750\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-180±\sqrt{32400+12\left(-750\right)}}{2\left(-3\right)}$$

Multiply 12 times -750.

$$x=\frac{-180±\sqrt{32400-9000}}{2\left(-3\right)}$$

Add 32400 to -9000.

$$x=\frac{-180±\sqrt{23400}}{2\left(-3\right)}$$

Take the square root of 23400.

$$x=\frac{-180±30\sqrt{26}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-180±30\sqrt{26}}{-6}$$

Now solve the equation x=\frac{-180±30\sqrt{26}}{-6} when ± is plus. Add -180 to 30\sqrt{26}.

$$x=\frac{30\sqrt{26}-180}{-6}$$

Divide -180+30\sqrt{26} by -6.

$$x=30-5\sqrt{26}$$

Now solve the equation x=\frac{-180±30\sqrt{26}}{-6} when ± is minus. Subtract 30\sqrt{26} from -180.

$$x=\frac{-30\sqrt{26}-180}{-6}$$

Divide -180-30\sqrt{26} by -6.

$$x=5\sqrt{26}+30$$

The equation is now solved.

$$x=30-5\sqrt{26} x=5\sqrt{26}+30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,50 since division by zero is not defined. Multiply both sides of the equation by x\left(x-50\right), the least common multiple of x,x-50.

$$\left(x-50\right)\times 15+x\times 15=3x\left(x-50\right)$$

Use the distributive property to multiply x-50 by 15.

$$15x-750+x\times 15=3x\left(x-50\right)$$

Combine 15x and x\times 15 to get 30x.

$$30x-750=3x\left(x-50\right)$$

Use the distributive property to multiply 3x by x-50.

$$30x-750=3x^{2}-150x$$

Subtract 3x^{2} from both sides.

$$30x-750-3x^{2}=-150x$$

Add 150x to both sides.

$$30x-750-3x^{2}+150x=0$$

Combine 30x and 150x to get 180x.

$$180x-750-3x^{2}=0$$

Add 750 to both sides. Anything plus zero gives itself.

$$180x-3x^{2}=750$$

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.

$$-3x^{2}+180x=750$$

Divide both sides by -3.

$$\frac{-3x^{2}+180x}{-3}=\frac{750}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{180}{-3}x=\frac{750}{-3}$$

Divide 180 by -3.

$$x^{2}-60x=\frac{750}{-3}$$

Divide 750 by -3.

$$x^{2}-60x=-250$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-250+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-250+900$$

Add -250 to 900.

$$x^{2}-60x+900=650$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=650$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{650}$$

Simplify.

$$x-30=5\sqrt{26} x-30=-5\sqrt{26}$$

Add 30 to both sides of the equation.

$$x=5\sqrt{26}+30 x=30-5\sqrt{26}$$