# SOLVE AN EQUATION? FIND A ZERO? YOUR CHOICE!

• PRACTICE (online exercises and printable worksheets)

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?

Master the ideas from this section