Solve 22500/x+50*(x-5)=22500

$$\frac{ 22500 }{ x } +50 \times (x-5)=22500$$

Solve for x

$$x = \frac{5 \sqrt{8209} + 455}{2} \approx 454.00882985$$

$$x=\frac{455-5\sqrt{8209}}{2}\approx 0.99117015$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$22500+50\left(x-5\right)x=22500x$$

Use the distributive property to multiply 50 by x-5.

$$22500+\left(50x-250\right)x=22500x$$

Use the distributive property to multiply 50x-250 by x.

$$22500+50x^{2}-250x=22500x$$

Subtract 22500x from both sides.

$$22500+50x^{2}-250x-22500x=0$$

Combine -250x and -22500x to get -22750x.

$$22500+50x^{2}-22750x=0$$

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.

$$50x^{2}-22750x+22500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 50 for a, -22750 for b, and 22500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22750\right)±\sqrt{\left(-22750\right)^{2}-4\times 50\times 22500}}{2\times 50}$$

Square -22750.

$$x=\frac{-\left(-22750\right)±\sqrt{517562500-4\times 50\times 22500}}{2\times 50}$$

Multiply -4 times 50.

$$x=\frac{-\left(-22750\right)±\sqrt{517562500-200\times 22500}}{2\times 50}$$

Multiply -200 times 22500.

$$x=\frac{-\left(-22750\right)±\sqrt{517562500-4500000}}{2\times 50}$$

Add 517562500 to -4500000.

$$x=\frac{-\left(-22750\right)±\sqrt{513062500}}{2\times 50}$$

Take the square root of 513062500.

$$x=\frac{-\left(-22750\right)±250\sqrt{8209}}{2\times 50}$$

The opposite of -22750 is 22750.

$$x=\frac{22750±250\sqrt{8209}}{2\times 50}$$

Multiply 2 times 50.

$$x=\frac{22750±250\sqrt{8209}}{100}$$

Now solve the equation x=\frac{22750±250\sqrt{8209}}{100} when ± is plus. Add 22750 to 250\sqrt{8209}.

$$x=\frac{250\sqrt{8209}+22750}{100}$$

Divide 22750+250\sqrt{8209} by 100.

$$x=\frac{5\sqrt{8209}+455}{2}$$

Now solve the equation x=\frac{22750±250\sqrt{8209}}{100} when ± is minus. Subtract 250\sqrt{8209} from 22750.

$$x=\frac{22750-250\sqrt{8209}}{100}$$

Divide 22750-250\sqrt{8209} by 100.

$$x=\frac{455-5\sqrt{8209}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{8209}+455}{2} x=\frac{455-5\sqrt{8209}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$22500+50\left(x-5\right)x=22500x$$

Use the distributive property to multiply 50 by x-5.

$$22500+\left(50x-250\right)x=22500x$$

Use the distributive property to multiply 50x-250 by x.

$$22500+50x^{2}-250x=22500x$$

Subtract 22500x from both sides.

$$22500+50x^{2}-250x-22500x=0$$

Combine -250x and -22500x to get -22750x.

$$22500+50x^{2}-22750x=0$$

Subtract 22500 from both sides. Anything subtracted from zero gives its negation.

$$50x^{2}-22750x=-22500$$

Divide both sides by 50.

$$\frac{50x^{2}-22750x}{50}=-\frac{22500}{50}$$

Dividing by 50 undoes the multiplication by 50.

$$x^{2}+\left(-\frac{22750}{50}\right)x=-\frac{22500}{50}$$

Divide -22750 by 50.

$$x^{2}-455x=-\frac{22500}{50}$$

Divide -22500 by 50.

$$x^{2}-455x=-450$$

Divide -455, the coefficient of the x term, by 2 to get -\frac{455}{2}. Then add the square of -\frac{455}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-455x+\left(-\frac{455}{2}\right)^{2}=-450+\left(-\frac{455}{2}\right)^{2}$$

Square -\frac{455}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-455x+\frac{207025}{4}=-450+\frac{207025}{4}$$

Add -450 to \frac{207025}{4}.

$$x^{2}-455x+\frac{207025}{4}=\frac{205225}{4}$$

Factor x^{2}-455x+\frac{207025}{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}.

$$\left(x-\frac{455}{2}\right)^{2}=\frac{205225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{455}{2}\right)^{2}}=\sqrt{\frac{205225}{4}}$$

Simplify.

$$x-\frac{455}{2}=\frac{5\sqrt{8209}}{2} x-\frac{455}{2}=-\frac{5\sqrt{8209}}{2}$$

Add \frac{455}{2} to both sides of the equation.

$$x=\frac{5\sqrt{8209}+455}{2} x=\frac{455-5\sqrt{8209}}{2}$$