Solve 6/x+6/x-15=6/300

$$\frac { 6 } { x } + \frac { 6 } { x - 15 } = \frac { 6 } { 300 }$$

Solve for x

$$x = \frac{15 \sqrt{1601} + 615}{2} \approx 607.593735356$$

$$x = \frac{615 - 15 \sqrt{1601}}{2} \approx 7.406264644$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,15 since division by zero is not defined. Multiply both sides of the equation by 300x\left(x-15\right), the least common multiple of x,x-15,300.

$$\left(300x-4500\right)\times 6+300x\times 6=x\left(x-15\right)\times 6$$

Use the distributive property to multiply 300x-4500 by 6.

$$1800x-27000+300x\times 6=x\left(x-15\right)\times 6$$

Multiply 300 and 6 to get 1800.

$$1800x-27000+1800x=x\left(x-15\right)\times 6$$

Combine 1800x and 1800x to get 3600x.

$$3600x-27000=x\left(x-15\right)\times 6$$

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

$$3600x-27000=\left(x^{2}-15x\right)\times 6$$

Use the distributive property to multiply x^{2}-15x by 6.

$$3600x-27000=6x^{2}-90x$$

Subtract 6x^{2} from both sides.

$$3600x-27000-6x^{2}=-90x$$

Add 90x to both sides.

$$3600x-27000-6x^{2}+90x=0$$

Combine 3600x and 90x to get 3690x.

$$3690x-27000-6x^{2}=0$$

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.

$$-6x^{2}+3690x-27000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 3690 for b, and -27000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3690±\sqrt{3690^{2}-4\left(-6\right)\left(-27000\right)}}{2\left(-6\right)}$$

Square 3690.

$$x=\frac{-3690±\sqrt{13616100-4\left(-6\right)\left(-27000\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-3690±\sqrt{13616100+24\left(-27000\right)}}{2\left(-6\right)}$$

Multiply 24 times -27000.

$$x=\frac{-3690±\sqrt{13616100-648000}}{2\left(-6\right)}$$

Add 13616100 to -648000.

$$x=\frac{-3690±\sqrt{12968100}}{2\left(-6\right)}$$

Take the square root of 12968100.

$$x=\frac{-3690±90\sqrt{1601}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-3690±90\sqrt{1601}}{-12}$$

Now solve the equation x=\frac{-3690±90\sqrt{1601}}{-12} when ± is plus. Add -3690 to 90\sqrt{1601}.

$$x=\frac{90\sqrt{1601}-3690}{-12}$$

Divide -3690+90\sqrt{1601} by -12.

$$x=\frac{615-15\sqrt{1601}}{2}$$

Now solve the equation x=\frac{-3690±90\sqrt{1601}}{-12} when ± is minus. Subtract 90\sqrt{1601} from -3690.

$$x=\frac{-90\sqrt{1601}-3690}{-12}$$

Divide -3690-90\sqrt{1601} by -12.

$$x=\frac{15\sqrt{1601}+615}{2}$$

The equation is now solved.

$$x=\frac{615-15\sqrt{1601}}{2} x=\frac{15\sqrt{1601}+615}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,15 since division by zero is not defined. Multiply both sides of the equation by 300x\left(x-15\right), the least common multiple of x,x-15,300.

$$\left(300x-4500\right)\times 6+300x\times 6=x\left(x-15\right)\times 6$$

Use the distributive property to multiply 300x-4500 by 6.

$$1800x-27000+300x\times 6=x\left(x-15\right)\times 6$$

Multiply 300 and 6 to get 1800.

$$1800x-27000+1800x=x\left(x-15\right)\times 6$$

Combine 1800x and 1800x to get 3600x.

$$3600x-27000=x\left(x-15\right)\times 6$$

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

$$3600x-27000=\left(x^{2}-15x\right)\times 6$$

Use the distributive property to multiply x^{2}-15x by 6.

$$3600x-27000=6x^{2}-90x$$

Subtract 6x^{2} from both sides.

$$3600x-27000-6x^{2}=-90x$$

Add 90x to both sides.

$$3600x-27000-6x^{2}+90x=0$$

Combine 3600x and 90x to get 3690x.

$$3690x-27000-6x^{2}=0$$

Add 27000 to both sides. Anything plus zero gives itself.

$$3690x-6x^{2}=27000$$

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.

$$-6x^{2}+3690x=27000$$

Divide both sides by -6.

$$\frac{-6x^{2}+3690x}{-6}=\frac{27000}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{3690}{-6}x=\frac{27000}{-6}$$

Divide 3690 by -6.

$$x^{2}-615x=\frac{27000}{-6}$$

Divide 27000 by -6.

$$x^{2}-615x=-4500$$

Divide -615, the coefficient of the x term, by 2 to get -\frac{615}{2}. Then add the square of -\frac{615}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-615x+\left(-\frac{615}{2}\right)^{2}=-4500+\left(-\frac{615}{2}\right)^{2}$$

Square -\frac{615}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-615x+\frac{378225}{4}=-4500+\frac{378225}{4}$$

Add -4500 to \frac{378225}{4}.

$$x^{2}-615x+\frac{378225}{4}=\frac{360225}{4}$$

Factor x^{2}-615x+\frac{378225}{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}.

$$\left(x-\frac{615}{2}\right)^{2}=\frac{360225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{615}{2}\right)^{2}}=\sqrt{\frac{360225}{4}}$$

Simplify.

$$x-\frac{615}{2}=\frac{15\sqrt{1601}}{2} x-\frac{615}{2}=-\frac{15\sqrt{1601}}{2}$$

Add \frac{615}{2} to both sides of the equation.

$$x=\frac{15\sqrt{1601}+615}{2} x=\frac{615-15\sqrt{1601}}{2}$$