Solve (2900-x-2500)(8+4*x/50)=5000

$$( 2900 - x - 2500 ) ( 8 + 4 \times \frac { x } { 50 } ) = 5000$$

Solve for x

$$x=150$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 50.

$$50\left(2900-x-2500\right)\left(8+4\times \frac{x}{50}\right)=250000$$

Subtract 2500 from 2900 to get 400.

$$50\left(400-x\right)\left(8+4\times \frac{x}{50}\right)=250000$$

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

$$50\left(400-x\right)\left(8+\frac{4x}{50}\right)=250000$$

Use the distributive property to multiply 50 by 400-x.

$$\left(20000-50x\right)\left(8+\frac{4x}{50}\right)=250000$$

Divide 4x by 50 to get \frac{2}{25}x.

$$\left(20000-50x\right)\left(8+\frac{2}{25}x\right)=250000$$

Apply the distributive property by multiplying each term of 20000-50x by each term of 8+\frac{2}{25}x.

$$160000+20000\times \frac{2}{25}x-400x-50x\times \frac{2}{25}x=250000$$

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

$$160000+20000\times \frac{2}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Express 20000\times \frac{2}{25} as a single fraction.

$$160000+\frac{20000\times 2}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Multiply 20000 and 2 to get 40000.

$$160000+\frac{40000}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Divide 40000 by 25 to get 1600.

$$160000+1600x-400x-50x^{2}\times \frac{2}{25}=250000$$

Combine 1600x and -400x to get 1200x.

$$160000+1200x-50x^{2}\times \frac{2}{25}=250000$$

Express -50\times \frac{2}{25} as a single fraction.

$$160000+1200x+\frac{-50\times 2}{25}x^{2}=250000$$

Multiply -50 and 2 to get -100.

$$160000+1200x+\frac{-100}{25}x^{2}=250000$$

Divide -100 by 25 to get -4.

$$160000+1200x-4x^{2}=250000$$

Subtract 250000 from both sides.

$$160000+1200x-4x^{2}-250000=0$$

Subtract 250000 from 160000 to get -90000.

$$-90000+1200x-4x^{2}=0$$

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.

$$-4x^{2}+1200x-90000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 1200 for b, and -90000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-4\right)\left(-90000\right)}}{2\left(-4\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-4\right)\left(-90000\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-1200±\sqrt{1440000+16\left(-90000\right)}}{2\left(-4\right)}$$

Multiply 16 times -90000.

$$x=\frac{-1200±\sqrt{1440000-1440000}}{2\left(-4\right)}$$

Add 1440000 to -1440000.

$$x=\frac{-1200±\sqrt{0}}{2\left(-4\right)}$$

Take the square root of 0.

$$x=-\frac{1200}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=-\frac{1200}{-8}$$

Divide -1200 by -8.

$$x=150$$

Steps for Completing the Square

Multiply both sides of the equation by 50.

$$50\left(2900-x-2500\right)\left(8+4\times \frac{x}{50}\right)=250000$$

Subtract 2500 from 2900 to get 400.

$$50\left(400-x\right)\left(8+4\times \frac{x}{50}\right)=250000$$

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

$$50\left(400-x\right)\left(8+\frac{4x}{50}\right)=250000$$

Use the distributive property to multiply 50 by 400-x.

$$\left(20000-50x\right)\left(8+\frac{4x}{50}\right)=250000$$

Divide 4x by 50 to get \frac{2}{25}x.

$$\left(20000-50x\right)\left(8+\frac{2}{25}x\right)=250000$$

Apply the distributive property by multiplying each term of 20000-50x by each term of 8+\frac{2}{25}x.

$$160000+20000\times \frac{2}{25}x-400x-50x\times \frac{2}{25}x=250000$$

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

$$160000+20000\times \frac{2}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Express 20000\times \frac{2}{25} as a single fraction.

$$160000+\frac{20000\times 2}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Multiply 20000 and 2 to get 40000.

$$160000+\frac{40000}{25}x-400x-50x^{2}\times \frac{2}{25}=250000$$

Divide 40000 by 25 to get 1600.

$$160000+1600x-400x-50x^{2}\times \frac{2}{25}=250000$$

Combine 1600x and -400x to get 1200x.

$$160000+1200x-50x^{2}\times \frac{2}{25}=250000$$

Express -50\times \frac{2}{25} as a single fraction.

$$160000+1200x+\frac{-50\times 2}{25}x^{2}=250000$$

Multiply -50 and 2 to get -100.

$$160000+1200x+\frac{-100}{25}x^{2}=250000$$

Divide -100 by 25 to get -4.

$$160000+1200x-4x^{2}=250000$$

Subtract 160000 from both sides.

$$1200x-4x^{2}=250000-160000$$

Subtract 160000 from 250000 to get 90000.

$$1200x-4x^{2}=90000$$

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.

$$-4x^{2}+1200x=90000$$

Divide both sides by -4.

$$\frac{-4x^{2}+1200x}{-4}=\frac{90000}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{1200}{-4}x=\frac{90000}{-4}$$

Divide 1200 by -4.

$$x^{2}-300x=\frac{90000}{-4}$$

Divide 90000 by -4.

$$x^{2}-300x=-22500$$

Divide -300, the coefficient of the x term, by 2 to get -150. Then add the square of -150 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-300x+\left(-150\right)^{2}=-22500+\left(-150\right)^{2}$$

Square -150.

$$x^{2}-300x+22500=-22500+22500$$

Add -22500 to 22500.

$$x^{2}-300x+22500=0$$

Factor x^{2}-300x+22500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-150\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-150=0 x-150=0$$

Add 150 to both sides of the equation.

$$x=150 x=150$$

The equation is now solved. Solutions are the same.

$$x=150$$