Solve x^2-421x+40000=0

$${ x }^{ 2 } -421x+40000 = 0$$

Solve for x

$$x = \frac{\sqrt{17241} + 421}{2} \approx 276.152494241$$

$$x = \frac{421 - \sqrt{17241}}{2} \approx 144.847505759$$

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.

$$x^{2}-421x+40000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -421 for b, and 40000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-421\right)±\sqrt{\left(-421\right)^{2}-4\times 40000}}{2}$$

Square -421.

$$x=\frac{-\left(-421\right)±\sqrt{177241-4\times 40000}}{2}$$

Multiply -4 times 40000.

$$x=\frac{-\left(-421\right)±\sqrt{177241-160000}}{2}$$

Add 177241 to -160000.

$$x=\frac{-\left(-421\right)±\sqrt{17241}}{2}$$

The opposite of -421 is 421.

$$x=\frac{421±\sqrt{17241}}{2}$$

Now solve the equation x=\frac{421±\sqrt{17241}}{2} when ± is plus. Add 421 to \sqrt{17241}.

$$x=\frac{\sqrt{17241}+421}{2}$$

Now solve the equation x=\frac{421±\sqrt{17241}}{2} when ± is minus. Subtract \sqrt{17241} from 421.

$$x=\frac{421-\sqrt{17241}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17241}+421}{2} x=\frac{421-\sqrt{17241}}{2}$$

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.

$$x^{2}-421x+40000=0$$

Subtract 40000 from both sides of the equation.

$$x^{2}-421x+40000-40000=-40000$$

Subtracting 40000 from itself leaves 0.

$$x^{2}-421x=-40000$$

Divide -421, the coefficient of the x term, by 2 to get -\frac{421}{2}. Then add the square of -\frac{421}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-421x+\left(-\frac{421}{2}\right)^{2}=-40000+\left(-\frac{421}{2}\right)^{2}$$

Square -\frac{421}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-421x+\frac{177241}{4}=-40000+\frac{177241}{4}$$

Add -40000 to \frac{177241}{4}.

$$x^{2}-421x+\frac{177241}{4}=\frac{17241}{4}$$

Factor x^{2}-421x+\frac{177241}{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}.

$$\left(x-\frac{421}{2}\right)^{2}=\frac{17241}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{421}{2}\right)^{2}}=\sqrt{\frac{17241}{4}}$$

Simplify.

$$x-\frac{421}{2}=\frac{\sqrt{17241}}{2} x-\frac{421}{2}=-\frac{\sqrt{17241}}{2}$$

Add \frac{421}{2} to both sides of the equation.

$$x=\frac{\sqrt{17241}+421}{2} x=\frac{421-\sqrt{17241}}{2}$$