# Roots of equation and probability

• MHB
• anemone
In summary, the roots of an equation are the values of the variable that satisfy the equation when substituted into it. They can be found using algebraic techniques or graphical techniques. Real roots are found on the real number line, while complex roots involve the imaginary number i. The roots of an equation are also the x-intercepts of its graph, and can represent possible outcomes in probability.
anemone
Gold Member
MHB
POTW Director
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Find the probability that $\sqrt{2+\sqrt{3}}\le |v+w|$.

We are given $1997$ equidistant points on the unit circle in the complex plane each of which is a solution to the equation $z^{1997}-1 = 0$. The angular spacing between neighbouring solutions is:

$\Delta \theta = \frac{2\pi}{1997} \approx 0,0031463$

WLOG we can let $u = 1 +i0$, and choose $w$ arbitrarily among the 1996 other possibilities. The angle, $\theta$, between $u$ and $w$, determines the absolute value of the sum $u + w$:

$|u+w| = 2 \cos \frac{\theta }{2} \geq \sqrt{2+\sqrt{3}}$

The low limit can be expressed as a cosine:

$\sqrt{2+\sqrt{3}} = \sqrt{2}\sqrt{1+\frac{\sqrt{3}}{2}} = \sqrt{2}\sqrt{\cos 0 + \cos \frac{\pi}{6}} = \sqrt{2}\sqrt{2 \cos^2 \frac{\pi}{12}} = 2 \cos \frac{\pi}{12}$

Hence, we have the condition: $\cos \frac{\theta}{2} \geq \cos \frac{\pi}{12}$ or $- \frac{\pi}{6}\leq \theta\leq \frac{\pi}{6}$

Thus, having chosen $u$, the $w$’s which fulfill the inequality may be chosen up to a max. angular distance of $\frac{\pi}{6}$ from $u$.

There are $\left \lfloor \frac{\pi}{6 \Delta \theta } \right \rfloor = 166$ solutions on both sides of $u$, which fulfill the inequality.

If $w$ is picked out at random, we get the probability

$\frac{2\cdot 166}{1996} = \frac{83}{499} \approx 16,63 \%.$

This result differs slightly from the fraction: $\frac{1}{6} = 16,666.. \%$, which would be obtained in the limit: $\Delta \theta \approx 0$ ($N \rightarrow \infty$).

## 1. What is the difference between a root of an equation and a solution of an equation?

A root of an equation is a value that makes the equation equal to zero, while a solution of an equation is any value that satisfies the equation. In other words, a root is a specific value that solves the equation, while a solution can be any value that makes the equation true.

## 2. How do you find the roots of a quadratic equation?

The roots of a quadratic equation can be found using the quadratic formula, which is (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0. Alternatively, you can also use factoring or completing the square methods to find the roots.

## 3. What is the relationship between roots of an equation and its graph?

The roots of an equation represent the x-intercepts of its graph, where the graph crosses the x-axis. This means that the values of x that make the equation equal to zero will also be the points where the graph intersects the x-axis.

## 4. How does probability relate to the roots of an equation?

The roots of an equation can be used to determine the probability of certain events occurring. For example, in a binomial distribution, the roots of the equation can be used to find the probability of a specific number of successes or failures in a given number of trials.

## 5. Can an equation have more than two roots?

Yes, an equation can have multiple roots, depending on its degree. For example, a quadratic equation can have up to two roots, while a cubic equation can have up to three roots. However, some equations may not have any real roots, or they may have complex roots.

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