Solve 2400/x-50/x+15=9

$$\frac { 2400 } { x } - \frac { 50 } { x + 15 } = 9$$

Solve for x

$$x = \frac{5 \sqrt{248089} + 2215}{18} \approx 261.412592793$$

$$x=\frac{2215-5\sqrt{248089}}{18}\approx -15.301481682$$

Steps Using the Quadratic Formula

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

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

Use the distributive property to multiply x+15 by 2400.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

Use the distributive property to multiply 9x by x+15.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Subtract 9x^{2} from both sides.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Subtract 135x from both sides.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

Combine 2400x and -135x to get 2265x.

$$2265x+36000-x\times 50-9x^{2}=0$$

Multiply -1 and 50 to get -50.

$$2265x+36000-50x-9x^{2}=0$$

Combine 2265x and -50x to get 2215x.

$$2215x+36000-9x^{2}=0$$

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.

$$-9x^{2}+2215x+36000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 2215 for b, and 36000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2215±\sqrt{2215^{2}-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

Square 2215.

$$x=\frac{-2215±\sqrt{4906225-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-2215±\sqrt{4906225+36\times 36000}}{2\left(-9\right)}$$

Multiply 36 times 36000.

$$x=\frac{-2215±\sqrt{4906225+1296000}}{2\left(-9\right)}$$

Add 4906225 to 1296000.

$$x=\frac{-2215±\sqrt{6202225}}{2\left(-9\right)}$$

Take the square root of 6202225.

$$x=\frac{-2215±5\sqrt{248089}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-2215±5\sqrt{248089}}{-18}$$

Now solve the equation x=\frac{-2215±5\sqrt{248089}}{-18} when ± is plus. Add -2215 to 5\sqrt{248089}.

$$x=\frac{5\sqrt{248089}-2215}{-18}$$

Divide -2215+5\sqrt{248089} by -18.

$$x=\frac{2215-5\sqrt{248089}}{18}$$

Now solve the equation x=\frac{-2215±5\sqrt{248089}}{-18} when ± is minus. Subtract 5\sqrt{248089} from -2215.

$$x=\frac{-5\sqrt{248089}-2215}{-18}$$

Divide -2215-5\sqrt{248089} by -18.

$$x=\frac{5\sqrt{248089}+2215}{18}$$

The equation is now solved.

$$x=\frac{2215-5\sqrt{248089}}{18} x=\frac{5\sqrt{248089}+2215}{18}$$

Steps for Completing the Square

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

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

Use the distributive property to multiply x+15 by 2400.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

Use the distributive property to multiply 9x by x+15.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Subtract 9x^{2} from both sides.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Subtract 135x from both sides.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

Combine 2400x and -135x to get 2265x.

$$2265x+36000-x\times 50-9x^{2}=0$$

Subtract 36000 from both sides. Anything subtracted from zero gives its negation.

$$2265x-x\times 50-9x^{2}=-36000$$

Multiply -1 and 50 to get -50.

$$2265x-50x-9x^{2}=-36000$$

Combine 2265x and -50x to get 2215x.

$$2215x-9x^{2}=-36000$$

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.

$$-9x^{2}+2215x=-36000$$

Divide both sides by -9.

$$\frac{-9x^{2}+2215x}{-9}=-\frac{36000}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{2215}{-9}x=-\frac{36000}{-9}$$

Divide 2215 by -9.

$$x^{2}-\frac{2215}{9}x=-\frac{36000}{-9}$$

Divide -36000 by -9.

$$x^{2}-\frac{2215}{9}x=4000$$

Divide -\frac{2215}{9}, the coefficient of the x term, by 2 to get -\frac{2215}{18}. Then add the square of -\frac{2215}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2215}{9}x+\left(-\frac{2215}{18}\right)^{2}=4000+\left(-\frac{2215}{18}\right)^{2}$$

Square -\frac{2215}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=4000+\frac{4906225}{324}$$

Add 4000 to \frac{4906225}{324}.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=\frac{6202225}{324}$$

Factor x^{2}-\frac{2215}{9}x+\frac{4906225}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2215}{18}\right)^{2}=\frac{6202225}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2215}{18}\right)^{2}}=\sqrt{\frac{6202225}{324}}$$

Simplify.

$$x-\frac{2215}{18}=\frac{5\sqrt{248089}}{18} x-\frac{2215}{18}=-\frac{5\sqrt{248089}}{18}$$

Add \frac{2215}{18} to both sides of the equation.

$$x=\frac{5\sqrt{248089}+2215}{18} x=\frac{2215-5\sqrt{248089}}{18}$$