Solve -2x^2+400x+25000=40000

$$- 2 x ^ { 2 } + 400 x + 25000 = 40000$$

Solve for x

$$x=50$$

$$x=150$$

Steps Using the Quadratic Formula

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.

$$-2x^{2}+400x+25000=40000$$

Subtract 40000 from both sides of the equation.

$$-2x^{2}+400x+25000-40000=40000-40000$$

Subtracting 40000 from itself leaves 0.

$$-2x^{2}+400x+25000-40000=0$$

Subtract 40000 from 25000.

$$-2x^{2}+400x-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 400 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\left(-2\right)\left(-15000\right)}}{2\left(-2\right)}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\left(-2\right)\left(-15000\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-400±\sqrt{160000+8\left(-15000\right)}}{2\left(-2\right)}$$

Multiply 8 times -15000.

$$x=\frac{-400±\sqrt{160000-120000}}{2\left(-2\right)}$$

Add 160000 to -120000.

$$x=\frac{-400±\sqrt{40000}}{2\left(-2\right)}$$

Take the square root of 40000.

$$x=\frac{-400±200}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-400±200}{-4}$$

Now solve the equation x=\frac{-400±200}{-4} when ± is plus. Add -400 to 200.

$$x=-\frac{200}{-4}$$

Divide -200 by -4.

$$x=50$$

Now solve the equation x=\frac{-400±200}{-4} when ± is minus. Subtract 200 from -400.

$$x=-\frac{600}{-4}$$

Divide -600 by -4.

$$x=150$$

The equation is now solved.

$$x=50 x=150$$

Steps for Completing the Square

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.

$$-2x^{2}+400x+25000=40000$$

Subtract 25000 from both sides of the equation.

$$-2x^{2}+400x+25000-25000=40000-25000$$

Subtracting 25000 from itself leaves 0.

$$-2x^{2}+400x=40000-25000$$

Subtract 25000 from 40000.

$$-2x^{2}+400x=15000$$

Divide both sides by -2.

$$\frac{-2x^{2}+400x}{-2}=\frac{15000}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{400}{-2}x=\frac{15000}{-2}$$

Divide 400 by -2.

$$x^{2}-200x=\frac{15000}{-2}$$

Divide 15000 by -2.

$$x^{2}-200x=-7500$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=-7500+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=-7500+10000$$

Add -7500 to 10000.

$$x^{2}-200x+10000=2500$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-100=50 x-100=-50$$

Add 100 to both sides of the equation.

$$x=150 x=50$$