Otherwise, I would lose the ability to say that they're equal. And now, we can square both sides of this equation.And so the left-hand side right over here simplifies to the principal square root of 5x plus 6. So we could square the principal square root of 5x plus 6 and we can square 9. Or we get 3 plus square root of 75 plus 6 is 81 needs to be equal to 12.
For most of this lesson, we'll be working with square roots.
For instance, this is a radical equation, because the variable is inside the square root: In general, we solve equations by isolating the variable; that is, we manipulate the equation to end up with the variable on one side of the "equals" sign, with a numerical value on the other side.
On the left-hand side of this equation, I have a square root. On the right-hand side, I've got a positive number.
Since both sides are known positive, squaring won't introduce extraneous solutions.
But I'll check my solution at the end, anyway, because the instructions require it.
First, I'll square both Because of this fact, my squaring of both sides of the equation will be an irreversible step.
When you do this-- when you square this, you get 5x plus 6. So we get x is equal to 15, but we need to make sure that this actually works for our original equation. And this is the principal root of 81 so it's positive 9.
If you square the square root of 5x plus 6, you're going to get 5x plus 6. On the left-hand side, we have 5x and on the right-hand side, we have 75. We get x is equal to-- let's see, it's 15, right? Maybe this would have worked if this was the negative square root. So it's 3 plus 9 needs to be equal to 12, which is absolutely true.
For instance, in my first example above, " Squaring both sides of an equation is an "irreversible" step, in the sense that, having taken the step, we can't necessarily go back to what we'd started with.
By squaring, we may have lost some of the original information.