Solve x^2-1548548x+200000000=0

$${ x }^{ 2 } -1548548x+200000000 = 0$$

Solve for x

$$x=2\sqrt{149825056769}+774274\approx 1548418.835980968$$

$$x=774274-2\sqrt{149825056769}\approx 129.164019032$$

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.

$$x^{2}-1548548x+200000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1548548 for b, and 200000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1548548\right)±\sqrt{\left(-1548548\right)^{2}-4\times 200000000}}{2}$$

Square -1548548.

$$x=\frac{-\left(-1548548\right)±\sqrt{2398000908304-4\times 200000000}}{2}$$

Multiply -4 times 200000000.

$$x=\frac{-\left(-1548548\right)±\sqrt{2398000908304-800000000}}{2}$$

Add 2398000908304 to -800000000.

$$x=\frac{-\left(-1548548\right)±\sqrt{2397200908304}}{2}$$

Take the square root of 2397200908304.

$$x=\frac{-\left(-1548548\right)±4\sqrt{149825056769}}{2}$$

The opposite of -1548548 is 1548548.

$$x=\frac{1548548±4\sqrt{149825056769}}{2}$$

Now solve the equation x=\frac{1548548±4\sqrt{149825056769}}{2} when ± is plus. Add 1548548 to 4\sqrt{149825056769}.

$$x=\frac{4\sqrt{149825056769}+1548548}{2}$$

Divide 1548548+4\sqrt{149825056769} by 2.

$$x=2\sqrt{149825056769}+774274$$

Now solve the equation x=\frac{1548548±4\sqrt{149825056769}}{2} when ± is minus. Subtract 4\sqrt{149825056769} from 1548548.

$$x=\frac{1548548-4\sqrt{149825056769}}{2}$$

Divide 1548548-4\sqrt{149825056769} by 2.

$$x=774274-2\sqrt{149825056769}$$

The equation is now solved.

$$x=2\sqrt{149825056769}+774274 x=774274-2\sqrt{149825056769}$$

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.

$$x^{2}-1548548x+200000000=0$$

Subtract 200000000 from both sides of the equation.

$$x^{2}-1548548x+200000000-200000000=-200000000$$

Subtracting 200000000 from itself leaves 0.

$$x^{2}-1548548x=-200000000$$

Divide -1548548, the coefficient of the x term, by 2 to get -774274. Then add the square of -774274 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1548548x+\left(-774274\right)^{2}=-200000000+\left(-774274\right)^{2}$$

Square -774274.

$$x^{2}-1548548x+599500227076=-200000000+599500227076$$

Add -200000000 to 599500227076.

$$x^{2}-1548548x+599500227076=599300227076$$

Factor x^{2}-1548548x+599500227076. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-774274\right)^{2}=599300227076$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-774274\right)^{2}}=\sqrt{599300227076}$$

Simplify.

$$x-774274=2\sqrt{149825056769} x-774274=-2\sqrt{149825056769}$$

Add 774274 to both sides of the equation.

$$x=2\sqrt{149825056769}+774274 x=774274-2\sqrt{149825056769}$$