Solve the equation 𝑥 squared
equals negative 16.
To solve an equation like this, we
do begin by solving as we would any equation with real solutions by performing a
series of inverse operations. In this case, we’re going to find
the square root of both sides of the equation. Before we do though, we choose to
rewrite negative 16 slightly. We’re going to write it as 16𝑖
squared. And we’ll see why we do this in a
moment. But for now, it works because 𝑖
squared is equal to negative one. And this means that 16𝑖 squared is
16 times negative one which is negative 16.
And now that we have the equation
𝑥 squared equals 16𝑖 squared, we can now find the square root of both sides of the
equation, remembering that we can take both the positive and negative root of 16𝑖
squared. The square root of 𝑥 squared is
𝑥. So 𝑥 is equal to the positive and
negative square root of 16𝑖 squared. And during this next step, it’s
going to become evident why we chose to write negative 16 as 16𝑖 squared. We can split the square root of
16𝑖 squared into the square root of 16 times the square root of 𝑖 squared.
The square root of 16 is four and
the square root of 𝑖 squared is simply 𝑖 so in turn we can see that 𝑥 is equal to
plus or minus four 𝑖. The solutions to the equation 𝑥
squared equals negative 16 are four 𝑖 and negative four 𝑖. And we should now be able to see
why we did write negative 16 as 16𝑖 squared. It made these final steps a little
easier to deal with.
And of course, we can check these
solutions by substituting them back into the original equation. Let’s try this for 𝑥 equals four
𝑖 first. 𝑥 squared is four 𝑖 squared. And of course, that’s four 𝑖 times
four 𝑖. Multiplication is commutative. It can be performed in any
order. So we can rewrite this as four
times four times 𝑖 times 𝑖. Four multiplied by four is 16 and
𝑖 times 𝑖 is 𝑖 squared. And since 𝑖 squared is negative
one, 𝑥 squared is negative 16 as required.
We can repeat this process for 𝑥
equals negative four 𝑖. 𝑥 squared is negative four 𝑖
times negative four 𝑖, which can in turn be written as negative four times negative
four times 𝑖 times 𝑖. Once again, since negative four
times negative four is positive 16, we get 16𝑖 squared which is negative 16 as