Solve x/8700+x*22500x=8000

$$\frac{ x }{ 8700+x } \times 22500x = 8000$$

Solve for x

$$x = \frac{16 \sqrt{24469} + 8}{45} \approx 55.795804876$$

$$x=\frac{8-16\sqrt{24469}}{45}\approx -55.44024932$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -8700 since division by zero is not defined. Multiply both sides of the equation by x+8700.

$$x\times 22500x=8000\left(x+8700\right)$$

Multiply x and x to get x^{2}.

$$x^{2}\times 22500=8000\left(x+8700\right)$$

Use the distributive property to multiply 8000 by x+8700.

$$x^{2}\times 22500=8000x+69600000$$

Subtract 8000x from both sides.

$$x^{2}\times 22500-8000x=69600000$$

Subtract 69600000 from both sides.

$$x^{2}\times 22500-8000x-69600000=0$$

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.

$$22500x^{2}-8000x-69600000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 22500 for a, -8000 for b, and -69600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8000\right)±\sqrt{\left(-8000\right)^{2}-4\times 22500\left(-69600000\right)}}{2\times 22500}$$

Square -8000.

$$x=\frac{-\left(-8000\right)±\sqrt{64000000-4\times 22500\left(-69600000\right)}}{2\times 22500}$$

Multiply -4 times 22500.

$$x=\frac{-\left(-8000\right)±\sqrt{64000000-90000\left(-69600000\right)}}{2\times 22500}$$

Multiply -90000 times -69600000.

$$x=\frac{-\left(-8000\right)±\sqrt{64000000+6264000000000}}{2\times 22500}$$

Add 64000000 to 6264000000000.

$$x=\frac{-\left(-8000\right)±\sqrt{6264064000000}}{2\times 22500}$$

Take the square root of 6264064000000.

$$x=\frac{-\left(-8000\right)±16000\sqrt{24469}}{2\times 22500}$$

The opposite of -8000 is 8000.

$$x=\frac{8000±16000\sqrt{24469}}{2\times 22500}$$

Multiply 2 times 22500.

$$x=\frac{8000±16000\sqrt{24469}}{45000}$$

Now solve the equation x=\frac{8000±16000\sqrt{24469}}{45000} when ± is plus. Add 8000 to 16000\sqrt{24469}.

$$x=\frac{16000\sqrt{24469}+8000}{45000}$$

Divide 8000+16000\sqrt{24469} by 45000.

$$x=\frac{16\sqrt{24469}+8}{45}$$

Now solve the equation x=\frac{8000±16000\sqrt{24469}}{45000} when ± is minus. Subtract 16000\sqrt{24469} from 8000.

$$x=\frac{8000-16000\sqrt{24469}}{45000}$$

Divide 8000-16000\sqrt{24469} by 45000.

$$x=\frac{8-16\sqrt{24469}}{45}$$

The equation is now solved.

$$x=\frac{16\sqrt{24469}+8}{45} x=\frac{8-16\sqrt{24469}}{45}$$

Steps for Completing the Square

Variable x cannot be equal to -8700 since division by zero is not defined. Multiply both sides of the equation by x+8700.

$$x\times 22500x=8000\left(x+8700\right)$$

Multiply x and x to get x^{2}.

$$x^{2}\times 22500=8000\left(x+8700\right)$$

Use the distributive property to multiply 8000 by x+8700.

$$x^{2}\times 22500=8000x+69600000$$

Subtract 8000x from both sides.

$$x^{2}\times 22500-8000x=69600000$$

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.

$$22500x^{2}-8000x=69600000$$

Divide both sides by 22500.

$$\frac{22500x^{2}-8000x}{22500}=\frac{69600000}{22500}$$

Dividing by 22500 undoes the multiplication by 22500.

$$x^{2}+\left(-\frac{8000}{22500}\right)x=\frac{69600000}{22500}$$

Reduce the fraction \frac{-8000}{22500} to lowest terms by extracting and canceling out 500.

$$x^{2}-\frac{16}{45}x=\frac{69600000}{22500}$$

Reduce the fraction \frac{69600000}{22500} to lowest terms by extracting and canceling out 7500.

$$x^{2}-\frac{16}{45}x=\frac{9280}{3}$$

Divide -\frac{16}{45}, the coefficient of the x term, by 2 to get -\frac{8}{45}. Then add the square of -\frac{8}{45} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{45}x+\left(-\frac{8}{45}\right)^{2}=\frac{9280}{3}+\left(-\frac{8}{45}\right)^{2}$$

Square -\frac{8}{45} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{45}x+\frac{64}{2025}=\frac{9280}{3}+\frac{64}{2025}$$

Add \frac{9280}{3} to \frac{64}{2025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{45}x+\frac{64}{2025}=\frac{6264064}{2025}$$

Factor x^{2}-\frac{16}{45}x+\frac{64}{2025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{45}\right)^{2}=\frac{6264064}{2025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{45}\right)^{2}}=\sqrt{\frac{6264064}{2025}}$$

Simplify.

$$x-\frac{8}{45}=\frac{16\sqrt{24469}}{45} x-\frac{8}{45}=-\frac{16\sqrt{24469}}{45}$$

Add \frac{8}{45} to both sides of the equation.

$$x=\frac{16\sqrt{24469}+8}{45} x=\frac{8-16\sqrt{24469}}{45}$$