Solve 1+x^2-21x=050565

$$1 + x ^ { 2 } - 21 x = 050565$$

Solve for x

$$x = \frac{\sqrt{437} + 21}{2} \approx 20.95227248$$

$$x=\frac{21-\sqrt{437}}{2}\approx 0.04772752$$

Steps Using the Quadratic Formula

Multiply 0 and 50565 to get 0.

$$1+x^{2}-21x=0$$

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.

$$x^{2}-21x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -21 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4}}{2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4}}{2}$$

Add 441 to -4.

$$x=\frac{-\left(-21\right)±\sqrt{437}}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{437}}{2}$$

Now solve the equation x=\frac{21±\sqrt{437}}{2} when ± is plus. Add 21 to \sqrt{437}.

$$x=\frac{\sqrt{437}+21}{2}$$

Now solve the equation x=\frac{21±\sqrt{437}}{2} when ± is minus. Subtract \sqrt{437} from 21.

$$x=\frac{21-\sqrt{437}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{437}+21}{2} x=\frac{21-\sqrt{437}}{2}$$

Steps for Completing the Square

Multiply 0 and 50565 to get 0.

$$1+x^{2}-21x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-21x=-1$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-1+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-1+\frac{441}{4}$$

Add -1 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{437}{4}$$

Factor x^{2}-21x+\frac{441}{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}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{437}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{437}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{\sqrt{437}}{2} x-\frac{21}{2}=-\frac{\sqrt{437}}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=\frac{\sqrt{437}+21}{2} x=\frac{21-\sqrt{437}}{2}$$