﻿ Solve an Equation? Find a Zero? Your Choice!

# Solve an Equation? Find a Zero? Your Choice!

In this section, ‘equation’ refers to an equation in one variable, and ‘function’ refers to a function that takes a single input (a number) and gives a single output (a number).

## Solve an Equation ...

Suppose you're asked to solve the equation: $$x^2 = -1$$

You'll need some clarification:

• Do you want only real-number solutions? If so, there aren't any—no real number, when squared, equals negative one.
• Are you allowing any complex number to be a solution? Then, there are two solutions, $\,i\,$ and $\,-i\,.$

## Find a Zero ...

Now, suppose you're asked to find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$

• That is, you want inputs whose output is zero.
• That is, you want values of $\,x\,$ for which $\,f(x) = 0\,.$
• That is, you want values of $\,x\,$ for which $\,x^2 + 1 = 0\,.$
• That is, you want values of $\,x\,$ for which $\,x^2 = -1\,.$

Now, you'll need the same clarification as above.

Notice something?

Every request to solve an equation can be rephrased as a request to find the zeros of a function, as follows:

• Get an equivalent equation with zero on the right-hand side.
• Define the function using the left-hand side of the equation.

For example:

 Initial request: Solve the equation $\,x^2 = -1\,.$ Get an equivalent equation with zero on the right-hand side: $x^2 + 1 = 0$ Define a function $\,f\,$ using the left-hand side of the equation: $f(x) = x^2 + 1$ Rephrase the initial request: Find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$

Every request to find the zeros of a function can be rephrased as a request to solve an equation, as follows:

• Set the output from the function equal to zero.
• Solve the resulting equation.

For example:

 Initial request: Find the zeros of the function $\,f\,$ defined by $\,f(x) = x^2 + 1\,.$ Set $\,f(x)\,$ equal to zero: $x^2 + 1 = 0$ Rephrase the initial request: Solve the equation $\,x^2 + 1 = 0\,.$

In both cases, you'll need clarification. Are you wanting only real number solutions/zeros? Or, are you allowing any complex number for the solutions/zeros?