Solve 10*200+left(10-xright)left(200+50xright)+4left(200-50xright)-3600=1250

$$10 \cdot 200+ \left( 10-x \right) \left( 200+50x \right) +4 \left( 200-50x \right) -3600 = 1250$$

Solve for x

$$x=1$$

Steps Using the Quadratic Formula

Multiply 10 and 200 to get 2000.

$$2000+\left(10-x\right)\left(200+50x\right)+4\left(200-50x\right)-3600=1250$$

Use the distributive property to multiply 10-x by 200+50x and combine like terms.

$$2000+2000+300x-50x^{2}+4\left(200-50x\right)-3600=1250$$

Add 2000 and 2000 to get 4000.

$$4000+300x-50x^{2}+4\left(200-50x\right)-3600=1250$$

Use the distributive property to multiply 4 by 200-50x.

$$4000+300x-50x^{2}+800-200x-3600=1250$$

Add 4000 and 800 to get 4800.

$$4800+300x-50x^{2}-200x-3600=1250$$

Combine 300x and -200x to get 100x.

$$4800+100x-50x^{2}-3600=1250$$

Subtract 3600 from 4800 to get 1200.

$$1200+100x-50x^{2}=1250$$

Subtract 1250 from both sides.

$$1200+100x-50x^{2}-1250=0$$

Subtract 1250 from 1200 to get -50.

$$-50+100x-50x^{2}=0$$

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.

$$-50x^{2}+100x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 100 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-50\right)\left(-50\right)}}{2\left(-50\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-50\right)\left(-50\right)}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-100±\sqrt{10000+200\left(-50\right)}}{2\left(-50\right)}$$

Multiply 200 times -50.

$$x=\frac{-100±\sqrt{10000-10000}}{2\left(-50\right)}$$

Add 10000 to -10000.

$$x=\frac{-100±\sqrt{0}}{2\left(-50\right)}$$

Take the square root of 0.

$$x=-\frac{100}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=-\frac{100}{-100}$$

Divide -100 by -100.

$$x=1$$

Steps for Completing the Square

Multiply 10 and 200 to get 2000.

$$2000+\left(10-x\right)\left(200+50x\right)+4\left(200-50x\right)-3600=1250$$

Use the distributive property to multiply 10-x by 200+50x and combine like terms.

$$2000+2000+300x-50x^{2}+4\left(200-50x\right)-3600=1250$$

Add 2000 and 2000 to get 4000.

$$4000+300x-50x^{2}+4\left(200-50x\right)-3600=1250$$

Use the distributive property to multiply 4 by 200-50x.

$$4000+300x-50x^{2}+800-200x-3600=1250$$

Add 4000 and 800 to get 4800.

$$4800+300x-50x^{2}-200x-3600=1250$$

Combine 300x and -200x to get 100x.

$$4800+100x-50x^{2}-3600=1250$$

Subtract 3600 from 4800 to get 1200.

$$1200+100x-50x^{2}=1250$$

Subtract 1200 from both sides.

$$100x-50x^{2}=1250-1200$$

Subtract 1200 from 1250 to get 50.

$$100x-50x^{2}=50$$

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.

$$-50x^{2}+100x=50$$

Divide both sides by -50.

$$\frac{-50x^{2}+100x}{-50}=\frac{50}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{100}{-50}x=\frac{50}{-50}$$

Divide 100 by -50.

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

Divide 50 by -50.

$$x^{2}-2x=-1$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-1+1$$

Add -1 to 1.

$$x^{2}-2x+1=0$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-1=0 x-1=0$$

Add 1 to both sides of the equation.

$$x=1 x=1$$

The equation is now solved. Solutions are the same.

$$x=1$$