Solve 600/x-10-600/x=3

$$\frac { 600 } { x - 10 } - \frac { 600 } { x } = 3$$

Solve for x

$$x=-40$$

$$x=50$$

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-10,x.

$$x\times 600-\left(x-10\right)\times 600=3x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 600.

$$x\times 600-\left(600x-6000\right)=3x\left(x-10\right)$$

To find the opposite of 600x-6000, find the opposite of each term.

$$x\times 600-600x+6000=3x\left(x-10\right)$$

Combine x\times 600 and -600x to get 0.

$$6000=3x\left(x-10\right)$$

Use the distributive property to multiply 3x by x-10.

$$6000=3x^{2}-30x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}-30x=6000$$

Subtract 6000 from both sides.

$$3x^{2}-30x-6000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -30 for b, and -6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 3\left(-6000\right)}}{2\times 3}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 3\left(-6000\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-30\right)±\sqrt{900-12\left(-6000\right)}}{2\times 3}$$

Multiply -12 times -6000.

$$x=\frac{-\left(-30\right)±\sqrt{900+72000}}{2\times 3}$$

Add 900 to 72000.

$$x=\frac{-\left(-30\right)±\sqrt{72900}}{2\times 3}$$

Take the square root of 72900.

$$x=\frac{-\left(-30\right)±270}{2\times 3}$$

The opposite of -30 is 30.

$$x=\frac{30±270}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{30±270}{6}$$

Now solve the equation x=\frac{30±270}{6} when ± is plus. Add 30 to 270.

$$x=\frac{300}{6}$$

Divide 300 by 6.

$$x=50$$

Now solve the equation x=\frac{30±270}{6} when ± is minus. Subtract 270 from 30.

$$x=-\frac{240}{6}$$

Divide -240 by 6.

$$x=-40$$

The equation is now solved.

$$x=50 x=-40$$

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-10,x.

$$x\times 600-\left(x-10\right)\times 600=3x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 600.

$$x\times 600-\left(600x-6000\right)=3x\left(x-10\right)$$

To find the opposite of 600x-6000, find the opposite of each term.

$$x\times 600-600x+6000=3x\left(x-10\right)$$

Combine x\times 600 and -600x to get 0.

$$6000=3x\left(x-10\right)$$

Use the distributive property to multiply 3x by x-10.

$$6000=3x^{2}-30x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}-30x=6000$$

Divide both sides by 3.

$$\frac{3x^{2}-30x}{3}=\frac{6000}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{30}{3}\right)x=\frac{6000}{3}$$

Divide -30 by 3.

$$x^{2}-10x=\frac{6000}{3}$$

Divide 6000 by 3.

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

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.

$$x^{2}-10x+\left(-5\right)^{2}=2000+\left(-5\right)^{2}$$

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.

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

Add 5 to both sides of the equation.

$$x=50 x=-40$$