Solve x^2-400x+37500=0

$${ x }^{ 2 } -400x+37500=0$$

Solve for x

$$x=150$$

$$x=250$$

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.

$$x^{2}-400x+37500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -400 for b, and 37500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\times 37500}}{2}$$

Square -400.

$$x=\frac{-\left(-400\right)±\sqrt{160000-4\times 37500}}{2}$$

Multiply -4 times 37500.

$$x=\frac{-\left(-400\right)±\sqrt{160000-150000}}{2}$$

Add 160000 to -150000.

$$x=\frac{-\left(-400\right)±\sqrt{10000}}{2}$$

Take the square root of 10000.

$$x=\frac{-\left(-400\right)±100}{2}$$

The opposite of -400 is 400.

$$x=\frac{400±100}{2}$$

Now solve the equation x=\frac{400±100}{2} when ± is plus. Add 400 to 100.

$$x=\frac{500}{2}$$

Divide 500 by 2.

$$x=250$$

Now solve the equation x=\frac{400±100}{2} when ± is minus. Subtract 100 from 400.

$$x=\frac{300}{2}$$

Divide 300 by 2.

$$x=150$$

The equation is now solved.

$$x=250 x=150$$

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.

$$x^{2}-400x+37500=0$$

Subtract 37500 from both sides of the equation.

$$x^{2}-400x+37500-37500=-37500$$

Subtracting 37500 from itself leaves 0.

$$x^{2}-400x=-37500$$

Divide -400, the coefficient of the x term, by 2 to get -200. Then add the square of -200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-400x+\left(-200\right)^{2}=-37500+\left(-200\right)^{2}$$

Square -200.

$$x^{2}-400x+40000=-37500+40000$$

Add -37500 to 40000.

$$x^{2}-400x+40000=2500$$

Factor x^{2}-400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-200\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-200\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-200=50 x-200=-50$$

Add 200 to both sides of the equation.

$$x=250 x=150$$