Hey everyone, it's Justin again. You've already learned how to graph quadratics in one form and
we're almost ready to learn a second form, but before we can start on that second form you'll
need to know how to use square roots to solve equations because it'll have a major role in the
second quadratic form. At the end of this lesson, you will be able to solve equations by using
square roots. First, we'll quickly review square roots, then practice together with three examples
of how to use square roots to solve equations before I give you one example to do on your own.
First, recall what a square root is. The square root of a value is the number that
you could square to make it. If 4 times 4 makes 16 then a √16 is 4. Squaring and square roots
are inverse operations that cancel each other out.
So now that that's out of the way, how would we solve this equation? x² = 25. To cancel out the
squared, we perform its inverse operation, square root. And whatever we do to one side has to happen
to the other side. The square and square root cancel out. All we need to do is simplify the
√25. You know what the √25? Is it's 5! Oh, right. I always forget about that. So that symbol is
called a plus-minus and that means that there are two values here, the positive version and the
negative version. To understand why, let's rewind a second. We said that since 4² is 16, the √16 is
4. That's only half right. There's another number that squares to make 16, -4² , that's also 16!
Multiplying negative times negative makes a positive just like multiplying two positives,
so then 16 doesn't have one square root, it has two! 4 or -4.
So that's why we got that plus-minus symbol, any positive number could be the result of squaring
a positive or a negative. Most of the time we have no way of knowing which is correct so we
just have to accept both answers as valid. We can even use this new symbol in our answer. Instead
of saying the answer is positive 5 or -5, you can just combine them and say positive or negative 5.
So then positive values always have two square roots, what about negative values?
What could we square to get negative 9? Well -3 times itself would still be positive 9. Positive 3
times itself also just makes positive 9. We can't multiply positive 3 and -3 because then we're not
squaring either number, they're different numbers. Is there even an answer? To put it simply,
no. There is no real number that can be squared to make -9 or any other negative number. We call this
undefined. The square root of any negative number will be undefined.
This video isn't just about square roots though, it's about using them to solve equations.
Let's try this example (x - 2)² = 100. It would be great to get that minus two moved over but
it's trapped inside the parentheses which are there because of the exponent. So in
order to solve this we'll have to deal with the square first. Cancel it out,
we throw a square root around it, do the same to the other side, square root cancels out the square
so now we've got an easy x - 2 on the left side. Right square root of 100 is 10,
except not necessarily. It could be positive or -10. We need to add 2 to both sides in order to
finish solving the equation, but how can we do that if we don't know whether the right side
is 10 or -10? If it's 10 and we add 2, our answer becomes 12, but if it's -10 and we add 2, we get
-8 instead. So which one is it? Well it's easy, it's both or either. It could it could be one
or the other and without any context here for what these numbers represent, all we can say
is that both answers satisfy the equation. If you want proof just plug the answers back in. For -8.
-8 - 2 is -10² is 100. So -8 works. What about 12?
Subtract 2 and we get 10, that squared is 100 so both -8 and 12 are correct solutions here.
How about an even harder one? You might be tempted here to just start with a big square root here,
after all that's how to cancel a square right? But here's the problem, this 25 is on the outside of
the square so it blocks the square root and square from affecting each other. I'll have to move
the 25 first. So we add 25 to both sides of the equation and now there is nothing
blocking the square and the square root from canceling out. On the right side, remember the
square root of a positive can be positive or negative so x + 3 is either equal to -5 or 5.
If it's -5, then we subtract 3 and get that x is -8. If it's positive 5,
then we subtract 3 and get that x is 2. Check, plug them into the original equation. If x is -8
then negative eight plus three is negative five -5² is 25 and 25 - 25 is 0. -8 works.
We try 2 instead, 2 + 3 is 5² is 25 - 25 is 0 so 2 works as well.
Did you happen to notice what's special about these two x values? when the input is -8 or 2,
the output 0. Why does that sound familiar? Because that's the exact definition of a
function's roots! When we set up our equation to solve, we had a quadratic function on one side
equal to 0 on the right side. When you solve equations with this setup you're solving for
the root of the quadratic function. Time to test what you just learned. What are the roots of this
function? Don't forget, finding the root means solving for when the function is equal to 0.
Pause the video while you work it out and then I'll give you the answer.
If we want to find the roots, we have to remember what that means. What values of x
cause the function to output 0? To solve this, we add 121 to both sides,
take the square root of both sides which will cancel out the square,
and split the equation in two. Either x - 1 = -11 or it equals positive 11 which means x
has to be either -10 or 12. Don't take my word for it though, check these answers yourself.
Throughout this video, you've actually been looking at the next
form of quadratic functions that you'll be learning about. In the next video,
you'll learn how this new form works and how to use it to graph quadratics. See you then!