Solve left(x-2500right)left(8+4900-x/50*4right)=5000

$$\left( x-2500 \right) \left( 8+ \frac{ 4900-x }{ 50 } \times 4 \right) = 5000$$

Solve for x

$$x=500\sqrt{6}+3750\approx 4974.744871392$$

$$x=3750-500\sqrt{6}\approx 2525.255128608$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 50.

$$50\left(x-2500\right)\left(8+\frac{4900-x}{50}\times 4\right)=250000$$

Express \frac{4900-x}{50}\times 4 as a single fraction.

$$50\left(x-2500\right)\left(8+\frac{\left(4900-x\right)\times 4}{50}\right)=250000$$

Use the distributive property to multiply 50 by x-2500.

$$\left(50x-125000\right)\left(8+\frac{\left(4900-x\right)\times 4}{50}\right)=250000$$

Use the distributive property to multiply 4900-x by 4.

$$\left(50x-125000\right)\left(8+\frac{19600-4x}{50}\right)=250000$$

Apply the distributive property by multiplying each term of 50x-125000 by each term of 8+\frac{19600-4x}{50}.

$$400x+50x\times \frac{19600-4x}{50}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Express 50\times \frac{19600-4x}{50} as a single fraction.

$$400x+\frac{50\left(19600-4x\right)}{50}x-1000000-125000\times \frac{19600-4x}{50}=250000$$

Cancel out 50 and 50.

$$400x+\left(19600-4x\right)x-1000000-125000\times \frac{19600-4x}{50}=250000$$

Use the distributive property to multiply 19600-4x by x.

$$400x+19600x-4x^{2}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Combine 400x and 19600x to get 20000x.

$$20000x-4x^{2}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Cancel out 50, the greatest common factor in 125000 and 50.

$$20000x-4x^{2}-1000000-2500\left(19600-4x\right)=250000$$

Use the distributive property to multiply -2500 by 19600-4x.

$$20000x-4x^{2}-1000000-49000000+10000x=250000$$

Subtract 49000000 from -1000000 to get -50000000.

$$20000x-4x^{2}-50000000+10000x=250000$$

Combine 20000x and 10000x to get 30000x.

$$30000x-4x^{2}-50000000=250000$$

Subtract 250000 from both sides.

$$30000x-4x^{2}-50000000-250000=0$$

Subtract 250000 from -50000000 to get -50250000.

$$30000x-4x^{2}-50250000=0$$

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.

$$-4x^{2}+30000x-50250000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 30000 for b, and -50250000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30000±\sqrt{30000^{2}-4\left(-4\right)\left(-50250000\right)}}{2\left(-4\right)}$$

Square 30000.

$$x=\frac{-30000±\sqrt{900000000-4\left(-4\right)\left(-50250000\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-30000±\sqrt{900000000+16\left(-50250000\right)}}{2\left(-4\right)}$$

Multiply 16 times -50250000.

$$x=\frac{-30000±\sqrt{900000000-804000000}}{2\left(-4\right)}$$

Add 900000000 to -804000000.

$$x=\frac{-30000±\sqrt{96000000}}{2\left(-4\right)}$$

Take the square root of 96000000.

$$x=\frac{-30000±4000\sqrt{6}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-30000±4000\sqrt{6}}{-8}$$

Now solve the equation x=\frac{-30000±4000\sqrt{6}}{-8} when ± is plus. Add -30000 to 4000\sqrt{6}.

$$x=\frac{4000\sqrt{6}-30000}{-8}$$

Divide -30000+4000\sqrt{6} by -8.

$$x=3750-500\sqrt{6}$$

Now solve the equation x=\frac{-30000±4000\sqrt{6}}{-8} when ± is minus. Subtract 4000\sqrt{6} from -30000.

$$x=\frac{-4000\sqrt{6}-30000}{-8}$$

Divide -30000-4000\sqrt{6} by -8.

$$x=500\sqrt{6}+3750$$

The equation is now solved.

$$x=3750-500\sqrt{6} x=500\sqrt{6}+3750$$

Steps for Completing the Square

Multiply both sides of the equation by 50.

$$50\left(x-2500\right)\left(8+\frac{4900-x}{50}\times 4\right)=250000$$

Express \frac{4900-x}{50}\times 4 as a single fraction.

$$50\left(x-2500\right)\left(8+\frac{\left(4900-x\right)\times 4}{50}\right)=250000$$

Use the distributive property to multiply 50 by x-2500.

$$\left(50x-125000\right)\left(8+\frac{\left(4900-x\right)\times 4}{50}\right)=250000$$

Use the distributive property to multiply 4900-x by 4.

$$\left(50x-125000\right)\left(8+\frac{19600-4x}{50}\right)=250000$$

Apply the distributive property by multiplying each term of 50x-125000 by each term of 8+\frac{19600-4x}{50}.

$$400x+50x\times \frac{19600-4x}{50}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Express 50\times \frac{19600-4x}{50} as a single fraction.

$$400x+\frac{50\left(19600-4x\right)}{50}x-1000000-125000\times \frac{19600-4x}{50}=250000$$

Cancel out 50 and 50.

$$400x+\left(19600-4x\right)x-1000000-125000\times \frac{19600-4x}{50}=250000$$

Use the distributive property to multiply 19600-4x by x.

$$400x+19600x-4x^{2}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Combine 400x and 19600x to get 20000x.

$$20000x-4x^{2}-1000000-125000\times \frac{19600-4x}{50}=250000$$

Cancel out 50, the greatest common factor in 125000 and 50.

$$20000x-4x^{2}-1000000-2500\left(19600-4x\right)=250000$$

Use the distributive property to multiply -2500 by 19600-4x.

$$20000x-4x^{2}-1000000-49000000+10000x=250000$$

Subtract 49000000 from -1000000 to get -50000000.

$$20000x-4x^{2}-50000000+10000x=250000$$

Combine 20000x and 10000x to get 30000x.

$$30000x-4x^{2}-50000000=250000$$

Add 50000000 to both sides.

$$30000x-4x^{2}=250000+50000000$$

Add 250000 and 50000000 to get 50250000.

$$30000x-4x^{2}=50250000$$

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.

$$-4x^{2}+30000x=50250000$$

Divide both sides by -4.

$$\frac{-4x^{2}+30000x}{-4}=\frac{50250000}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{30000}{-4}x=\frac{50250000}{-4}$$

Divide 30000 by -4.

$$x^{2}-7500x=\frac{50250000}{-4}$$

Divide 50250000 by -4.

$$x^{2}-7500x=-12562500$$

Divide -7500, the coefficient of the x term, by 2 to get -3750. Then add the square of -3750 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7500x+\left(-3750\right)^{2}=-12562500+\left(-3750\right)^{2}$$

Square -3750.

$$x^{2}-7500x+14062500=-12562500+14062500$$

Add -12562500 to 14062500.

$$x^{2}-7500x+14062500=1500000$$

Factor x^{2}-7500x+14062500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3750\right)^{2}=1500000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3750\right)^{2}}=\sqrt{1500000}$$

Simplify.

$$x-3750=500\sqrt{6} x-3750=-500\sqrt{6}$$

Add 3750 to both sides of the equation.

$$x=500\sqrt{6}+3750 x=3750-500\sqrt{6}$$