Solve x^2-225x+400=0

$${ x }^{ 2 } -225x+400=0$$

Solve for x

$$x = \frac{5 \sqrt{1961} + 225}{2} \approx 223.207949127$$

$$x = \frac{225 - 5 \sqrt{1961}}{2} \approx 1.792050873$$

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.

$$x^{2}-225x+400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -225 for b, and 400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times 400}}{2}$$

Square -225.

$$x=\frac{-\left(-225\right)±\sqrt{50625-4\times 400}}{2}$$

Multiply -4 times 400.

$$x=\frac{-\left(-225\right)±\sqrt{50625-1600}}{2}$$

Add 50625 to -1600.

$$x=\frac{-\left(-225\right)±\sqrt{49025}}{2}$$

Take the square root of 49025.

$$x=\frac{-\left(-225\right)±5\sqrt{1961}}{2}$$

The opposite of -225 is 225.

$$x=\frac{225±5\sqrt{1961}}{2}$$

Now solve the equation x=\frac{225±5\sqrt{1961}}{2} when ± is plus. Add 225 to 5\sqrt{1961}.

$$x=\frac{5\sqrt{1961}+225}{2}$$

Now solve the equation x=\frac{225±5\sqrt{1961}}{2} when ± is minus. Subtract 5\sqrt{1961} from 225.

$$x=\frac{225-5\sqrt{1961}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{1961}+225}{2} x=\frac{225-5\sqrt{1961}}{2}$$

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.

$$x^{2}-225x+400=0$$

Subtract 400 from both sides of the equation.

$$x^{2}-225x+400-400=-400$$

Subtracting 400 from itself leaves 0.

$$x^{2}-225x=-400$$

Divide -225, the coefficient of the x term, by 2 to get -\frac{225}{2}. Then add the square of -\frac{225}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-225x+\left(-\frac{225}{2}\right)^{2}=-400+\left(-\frac{225}{2}\right)^{2}$$

Square -\frac{225}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-225x+\frac{50625}{4}=-400+\frac{50625}{4}$$

Add -400 to \frac{50625}{4}.

$$x^{2}-225x+\frac{50625}{4}=\frac{49025}{4}$$

Factor x^{2}-225x+\frac{50625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{225}{2}\right)^{2}=\frac{49025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{225}{2}\right)^{2}}=\sqrt{\frac{49025}{4}}$$

Simplify.

$$x-\frac{225}{2}=\frac{5\sqrt{1961}}{2} x-\frac{225}{2}=-\frac{5\sqrt{1961}}{2}$$

Add \frac{225}{2} to both sides of the equation.

$$x=\frac{5\sqrt{1961}+225}{2} x=\frac{225-5\sqrt{1961}}{2}$$