Solve x^2-1200x+40000=0

$${ x }^{ 2 } -1200x+40000 = 0$$

Solve for x

$$x=400\sqrt{2}+600\approx 1165.685424949$$

$$x=600-400\sqrt{2}\approx 34.314575051$$

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.

$$x^{2}-1200x+40000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1200 for b, and 40000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1200\right)±\sqrt{\left(-1200\right)^{2}-4\times 40000}}{2}$$

Square -1200.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-4\times 40000}}{2}$$

Multiply -4 times 40000.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-160000}}{2}$$

Add 1440000 to -160000.

$$x=\frac{-\left(-1200\right)±\sqrt{1280000}}{2}$$

Take the square root of 1280000.

$$x=\frac{-\left(-1200\right)±800\sqrt{2}}{2}$$

The opposite of -1200 is 1200.

$$x=\frac{1200±800\sqrt{2}}{2}$$

Now solve the equation x=\frac{1200±800\sqrt{2}}{2} when ± is plus. Add 1200 to 800\sqrt{2}.

$$x=\frac{800\sqrt{2}+1200}{2}$$

Divide 1200+800\sqrt{2} by 2.

$$x=400\sqrt{2}+600$$

Now solve the equation x=\frac{1200±800\sqrt{2}}{2} when ± is minus. Subtract 800\sqrt{2} from 1200.

$$x=\frac{1200-800\sqrt{2}}{2}$$

Divide 1200-800\sqrt{2} by 2.

$$x=600-400\sqrt{2}$$

The equation is now solved.

$$x=400\sqrt{2}+600 x=600-400\sqrt{2}$$

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.

$$x^{2}-1200x+40000=0$$

Subtract 40000 from both sides of the equation.

$$x^{2}-1200x+40000-40000=-40000$$

Subtracting 40000 from itself leaves 0.

$$x^{2}-1200x=-40000$$

Divide -1200, the coefficient of the x term, by 2 to get -600. Then add the square of -600 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1200x+\left(-600\right)^{2}=-40000+\left(-600\right)^{2}$$

Square -600.

$$x^{2}-1200x+360000=-40000+360000$$

Add -40000 to 360000.

$$x^{2}-1200x+360000=320000$$

Factor x^{2}-1200x+360000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-600\right)^{2}=320000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-600\right)^{2}}=\sqrt{320000}$$

Simplify.

$$x-600=400\sqrt{2} x-600=-400\sqrt{2}$$

Add 600 to both sides of the equation.

$$x=400\sqrt{2}+600 x=600-400\sqrt{2}$$