Solve (4-x)(200+5x)+4(200-50x)=1250

$$(4-x)(200+5x)+4(200-50x)=1250$$

Solve for x (complex solution)

$$x=\sqrt{1514}-38\approx 0.910152917$$

$$x=-\left(\sqrt{1514}+38\right)\approx -76.910152917$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4-x by 200+5x and combine like terms.

$$800-180x-5x^{2}+4\left(200-50x\right)=1250$$

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

$$800-180x-5x^{2}+800-200x=1250$$

Add 800 and 800 to get 1600.

$$1600-180x-5x^{2}-200x=1250$$

Combine -180x and -200x to get -380x.

$$1600-380x-5x^{2}=1250$$

Subtract 1250 from both sides.

$$1600-380x-5x^{2}-1250=0$$

Subtract 1250 from 1600 to get 350.

$$350-380x-5x^{2}=0$$

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.

$$-5x^{2}-380x+350=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -380 for b, and 350 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-380\right)±\sqrt{\left(-380\right)^{2}-4\left(-5\right)\times 350}}{2\left(-5\right)}$$

Square -380.

$$x=\frac{-\left(-380\right)±\sqrt{144400-4\left(-5\right)\times 350}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-380\right)±\sqrt{144400+20\times 350}}{2\left(-5\right)}$$

Multiply 20 times 350.

$$x=\frac{-\left(-380\right)±\sqrt{144400+7000}}{2\left(-5\right)}$$

Add 144400 to 7000.

$$x=\frac{-\left(-380\right)±\sqrt{151400}}{2\left(-5\right)}$$

Take the square root of 151400.

$$x=\frac{-\left(-380\right)±10\sqrt{1514}}{2\left(-5\right)}$$

The opposite of -380 is 380.

$$x=\frac{380±10\sqrt{1514}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{380±10\sqrt{1514}}{-10}$$

Now solve the equation x=\frac{380±10\sqrt{1514}}{-10} when ± is plus. Add 380 to 10\sqrt{1514}.

$$x=\frac{10\sqrt{1514}+380}{-10}$$

Divide 380+10\sqrt{1514} by -10.

$$x=-\left(\sqrt{1514}+38\right)$$

Now solve the equation x=\frac{380±10\sqrt{1514}}{-10} when ± is minus. Subtract 10\sqrt{1514} from 380.

$$x=\frac{380-10\sqrt{1514}}{-10}$$

Divide 380-10\sqrt{1514} by -10.

$$x=\sqrt{1514}-38$$

The equation is now solved.

$$x=-\left(\sqrt{1514}+38\right) x=\sqrt{1514}-38$$

Steps for Completing the Square

Use the distributive property to multiply 4-x by 200+5x and combine like terms.

$$800-180x-5x^{2}+4\left(200-50x\right)=1250$$

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

$$800-180x-5x^{2}+800-200x=1250$$

Add 800 and 800 to get 1600.

$$1600-180x-5x^{2}-200x=1250$$

Combine -180x and -200x to get -380x.

$$1600-380x-5x^{2}=1250$$

Subtract 1600 from both sides.

$$-380x-5x^{2}=1250-1600$$

Subtract 1600 from 1250 to get -350.

$$-380x-5x^{2}=-350$$

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.

$$-5x^{2}-380x=-350$$

Divide both sides by -5.

$$\frac{-5x^{2}-380x}{-5}=-\frac{350}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{380}{-5}\right)x=-\frac{350}{-5}$$

Divide -380 by -5.

$$x^{2}+76x=-\frac{350}{-5}$$

Divide -350 by -5.

$$x^{2}+76x=70$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=70+38^{2}$$

Square 38.

$$x^{2}+76x+1444=70+1444$$

Add 70 to 1444.

$$x^{2}+76x+1444=1514$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=1514$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{1514}$$

Simplify.

$$x+38=\sqrt{1514} x+38=-\sqrt{1514}$$

Subtract 38 from both sides of the equation.

$$x=\sqrt{1514}-38 x=-\sqrt{1514}-38$$

Solve for x

$$x=\sqrt{1514}-38\approx 0.910152917$$

$$x=-\sqrt{1514}-38\approx -76.910152917$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4-x by 200+5x and combine like terms.

$$800-180x-5x^{2}+4\left(200-50x\right)=1250$$

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

$$800-180x-5x^{2}+800-200x=1250$$

Add 800 and 800 to get 1600.

$$1600-180x-5x^{2}-200x=1250$$

Combine -180x and -200x to get -380x.

$$1600-380x-5x^{2}=1250$$

Subtract 1250 from both sides.

$$1600-380x-5x^{2}-1250=0$$

Subtract 1250 from 1600 to get 350.

$$350-380x-5x^{2}=0$$

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.

$$-5x^{2}-380x+350=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -380 for b, and 350 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-380\right)±\sqrt{\left(-380\right)^{2}-4\left(-5\right)\times 350}}{2\left(-5\right)}$$

Square -380.

$$x=\frac{-\left(-380\right)±\sqrt{144400-4\left(-5\right)\times 350}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-380\right)±\sqrt{144400+20\times 350}}{2\left(-5\right)}$$

Multiply 20 times 350.

$$x=\frac{-\left(-380\right)±\sqrt{144400+7000}}{2\left(-5\right)}$$

Add 144400 to 7000.

$$x=\frac{-\left(-380\right)±\sqrt{151400}}{2\left(-5\right)}$$

Take the square root of 151400.

$$x=\frac{-\left(-380\right)±10\sqrt{1514}}{2\left(-5\right)}$$

The opposite of -380 is 380.

$$x=\frac{380±10\sqrt{1514}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{380±10\sqrt{1514}}{-10}$$

Now solve the equation x=\frac{380±10\sqrt{1514}}{-10} when ± is plus. Add 380 to 10\sqrt{1514}.

$$x=\frac{10\sqrt{1514}+380}{-10}$$

Divide 380+10\sqrt{1514} by -10.

$$x=-\left(\sqrt{1514}+38\right)$$

Now solve the equation x=\frac{380±10\sqrt{1514}}{-10} when ± is minus. Subtract 10\sqrt{1514} from 380.

$$x=\frac{380-10\sqrt{1514}}{-10}$$

Divide 380-10\sqrt{1514} by -10.

$$x=\sqrt{1514}-38$$

The equation is now solved.

$$x=-\left(\sqrt{1514}+38\right) x=\sqrt{1514}-38$$

Steps for Completing the Square

Use the distributive property to multiply 4-x by 200+5x and combine like terms.

$$800-180x-5x^{2}+4\left(200-50x\right)=1250$$

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

$$800-180x-5x^{2}+800-200x=1250$$

Add 800 and 800 to get 1600.

$$1600-180x-5x^{2}-200x=1250$$

Combine -180x and -200x to get -380x.

$$1600-380x-5x^{2}=1250$$

Subtract 1600 from both sides.

$$-380x-5x^{2}=1250-1600$$

Subtract 1600 from 1250 to get -350.

$$-380x-5x^{2}=-350$$

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.

$$-5x^{2}-380x=-350$$

Divide both sides by -5.

$$\frac{-5x^{2}-380x}{-5}=-\frac{350}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{380}{-5}\right)x=-\frac{350}{-5}$$

Divide -380 by -5.

$$x^{2}+76x=-\frac{350}{-5}$$

Divide -350 by -5.

$$x^{2}+76x=70$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=70+38^{2}$$

Square 38.

$$x^{2}+76x+1444=70+1444$$

Add 70 to 1444.

$$x^{2}+76x+1444=1514$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=1514$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{1514}$$

Simplify.

$$x+38=\sqrt{1514} x+38=-\sqrt{1514}$$

Subtract 38 from both sides of the equation.

$$x=\sqrt{1514}-38 x=-\sqrt{1514}-38$$