Solve (1001/x)-(100/x+5)=2/3

$$( \frac { 1001 } { x } ) - ( \frac { 100 } { x + 5 } ) = \frac { 2 } { 3 }$$

Solve for x

$$x = \frac{\sqrt{7372369} + 2693}{4} \approx 1352.052668306$$

$$x=\frac{2693-\sqrt{7372369}}{4}\approx -5.552668306$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+5\right), the least common multiple of x,x+5,3.

$$\left(3x+15\right)\times 1001-3x\times 100=2x\left(x+5\right)$$

Use the distributive property to multiply 3x+15 by 1001.

$$3003x+15015-3x\times 100=2x\left(x+5\right)$$

Multiply 3 and 100 to get 300.

$$3003x+15015-300x=2x\left(x+5\right)$$

Use the distributive property to multiply 2x by x+5.

$$3003x+15015-300x=2x^{2}+10x$$

Subtract 2x^{2} from both sides.

$$3003x+15015-300x-2x^{2}=10x$$

Subtract 10x from both sides.

$$3003x+15015-300x-2x^{2}-10x=0$$

Combine 3003x and -10x to get 2993x.

$$2993x+15015-300x-2x^{2}=0$$

Combine 2993x and -300x to get 2693x.

$$2693x+15015-2x^{2}=0$$

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.

$$-2x^{2}+2693x+15015=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 2693 for b, and 15015 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2693±\sqrt{2693^{2}-4\left(-2\right)\times 15015}}{2\left(-2\right)}$$

Square 2693.

$$x=\frac{-2693±\sqrt{7252249-4\left(-2\right)\times 15015}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-2693±\sqrt{7252249+8\times 15015}}{2\left(-2\right)}$$

Multiply 8 times 15015.

$$x=\frac{-2693±\sqrt{7252249+120120}}{2\left(-2\right)}$$

Add 7252249 to 120120.

$$x=\frac{-2693±\sqrt{7372369}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-2693±\sqrt{7372369}}{-4}$$

Now solve the equation x=\frac{-2693±\sqrt{7372369}}{-4} when ± is plus. Add -2693 to \sqrt{7372369}.

$$x=\frac{\sqrt{7372369}-2693}{-4}$$

Divide -2693+\sqrt{7372369} by -4.

$$x=\frac{2693-\sqrt{7372369}}{4}$$

Now solve the equation x=\frac{-2693±\sqrt{7372369}}{-4} when ± is minus. Subtract \sqrt{7372369} from -2693.

$$x=\frac{-\sqrt{7372369}-2693}{-4}$$

Divide -2693-\sqrt{7372369} by -4.

$$x=\frac{\sqrt{7372369}+2693}{4}$$

The equation is now solved.

$$x=\frac{2693-\sqrt{7372369}}{4} x=\frac{\sqrt{7372369}+2693}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+5\right), the least common multiple of x,x+5,3.

$$\left(3x+15\right)\times 1001-3x\times 100=2x\left(x+5\right)$$

Use the distributive property to multiply 3x+15 by 1001.

$$3003x+15015-3x\times 100=2x\left(x+5\right)$$

Multiply 3 and 100 to get 300.

$$3003x+15015-300x=2x\left(x+5\right)$$

Use the distributive property to multiply 2x by x+5.

$$3003x+15015-300x=2x^{2}+10x$$

Subtract 2x^{2} from both sides.

$$3003x+15015-300x-2x^{2}=10x$$

Subtract 10x from both sides.

$$3003x+15015-300x-2x^{2}-10x=0$$

Combine 3003x and -10x to get 2993x.

$$2993x+15015-300x-2x^{2}=0$$

Subtract 15015 from both sides. Anything subtracted from zero gives its negation.

$$2993x-300x-2x^{2}=-15015$$

Combine 2993x and -300x to get 2693x.

$$2693x-2x^{2}=-15015$$

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.

$$-2x^{2}+2693x=-15015$$

Divide both sides by -2.

$$\frac{-2x^{2}+2693x}{-2}=-\frac{15015}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{2693}{-2}x=-\frac{15015}{-2}$$

Divide 2693 by -2.

$$x^{2}-\frac{2693}{2}x=-\frac{15015}{-2}$$

Divide -15015 by -2.

$$x^{2}-\frac{2693}{2}x=\frac{15015}{2}$$

Divide -\frac{2693}{2}, the coefficient of the x term, by 2 to get -\frac{2693}{4}. Then add the square of -\frac{2693}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2693}{2}x+\left(-\frac{2693}{4}\right)^{2}=\frac{15015}{2}+\left(-\frac{2693}{4}\right)^{2}$$

Square -\frac{2693}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2693}{2}x+\frac{7252249}{16}=\frac{15015}{2}+\frac{7252249}{16}$$

Add \frac{15015}{2} to \frac{7252249}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2693}{2}x+\frac{7252249}{16}=\frac{7372369}{16}$$

Factor x^{2}-\frac{2693}{2}x+\frac{7252249}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2693}{4}\right)^{2}=\frac{7372369}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2693}{4}\right)^{2}}=\sqrt{\frac{7372369}{16}}$$

Simplify.

$$x-\frac{2693}{4}=\frac{\sqrt{7372369}}{4} x-\frac{2693}{4}=-\frac{\sqrt{7372369}}{4}$$

Add \frac{2693}{4} to both sides of the equation.

$$x=\frac{\sqrt{7372369}+2693}{4} x=\frac{2693-\sqrt{7372369}}{4}$$