# Question Video: Solving Quadratic Equations with Imaginary Roots Mathematics • 12th Grade

Solve the equation π₯Β² = β16.

02:17

### Video Transcript

Solve the equation π₯ 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, weβre going to find the square root of both sides of the equation. Before we do though, we choose to rewrite negative 16 slightly. Weβre going to write it as 16π squared. And weβll see why we do this in a moment. But for now, it works because π squared is equal to negative one. And this means that 16π squared is 16 times negative one which is negative 16.

And now that we have the equation π₯ squared equals 16π 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 16π squared. The square root of π₯ squared is π₯. So π₯ is equal to the positive and negative square root of 16π squared. And during this next step, itβs going to become evident why we chose to write negative 16 as 16π squared. We can split the square root of 16π squared into the square root of 16 times the square root of π squared.

The square root of 16 is four and the square root of π squared is simply π so in turn we can see that π₯ is equal to plus or minus four π. The solutions to the equation π₯ squared equals negative 16 are four π and negative four π. And we should now be able to see why we did write negative 16 as 16π 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. Letβs try this for π₯ equals four π first. π₯ squared is four π squared. And of course, thatβs four π times four π. Multiplication is commutative. It can be performed in any order. So we can rewrite this as four times four times π times π. Four multiplied by four is 16 and π times π is π squared. And since π squared is negative one, π₯ squared is negative 16 as required.

We can repeat this process for π₯ equals negative four π. π₯ squared is negative four π times negative four π, which can in turn be written as negative four times negative four times π times π. Once again, since negative four times negative four is positive 16, we get 16π squared which is negative 16 as required.