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The Wizard of Oz: From Fractions to Formulas
Square Roots
SallyAnn Keizer
Hello and welcome again to another episode of Algebra Workout. I’m SallyAnn Keiser and
this is Alan Borthwick.
Alan Borthwick
We’re going to be looking at square roots, which are sometimes a bit tricky to deal with. We’ll
be showing you some techniques to try to ensure that you don’t make silly mistakes.
SallyAnn Keizer
Once again, I’ll be looking at some examples, and then Alan is going to go through some
general rules to help you. So, let’s get ready.
I’ll start by introducing you to one of those tricky issues. Look at this equation.
x = √(x + 2)
I’ll have a go at solving it. Suppose I start by squaring both sides. That gives me
x2 = x + 2
And bringing this all over to the left hand side, I have the quadratic
x2 – x – 2 = 0
Now I need to factorise that. And I can see that it’s
(x – 2) x (x + 1) = 0
So, I have my solutions.
x = 2, or x = -1
So, what happens if I substitute this back into the original equation? Well, when x = 2, that
gives me
2 = √ (2 + 2)
which is 4, which is 2, so that’s correct. But, when I substitute x = -1 back into the equation,
that gives me
-1 = √ (-1 + 2)
which is √+1, which is 1.
-1 is equal to +1. So, there’s something funny going on there. The solution, x = -1 doesn’t
work. You have to be careful when there are square roots around, and I’m going to hand over
to Alan to explain some basics.
Alan Borthwick
Thanks, SallyAnn. Okay, there’s a convention we use which says that where a is any positive
number or 0, then if √a = b, that means that b2 = a. Now, when we write the √a, we always
mean, by convention, the positive root, so b is also positive. Let’s look at some examples
using real numbers to try and clarify that. Suppose a is 1. Well, in that case, √a = b, 1, so
that 12 = 1. Let’s try another couple of examples. Say we have √4 = 2, well, that simply
means that 22 = 4. Another one, √81 is 9, so that means that 92 = 81. Now, one thing you
might have noticed here is that the square root of a number is smaller than the actual number.
Be careful, is that always the case?
SallyAnn Keizer
Well, what about the square roots of positive numbers less than 1, then? √1/4 is 1/2, because
1/2 2 is 1/4, which is different from these examples Alan’s used.
Alan Borthwick
Okay. Well, let’s have a look in some more detail at square roots of fractions and decimals.
What about if you consider 2/10 all squared? Well, in that case, you get 2 2/102, which is
4/100. Or, in decimal terms, 0.22 = 0.04. So, in this case, when you’re looking at numbers
less than 1, squaring gives a smaller number. Now, making the link to the left hand side of
the screen, the square root of 0.04 = 0.2. Right, let’s do the same sort of thing for 81 and 9.
Now, nine tenths all squared is 81/100, or, in decimal terms, 0.9 2 is 0.81. And you can now
say that √0.81 is 0.9, which is again, larger, than 0.81.
SallyAnn Keizer
Okay. Remember, when you use the square root sign, then the answer is always positive or
zero, it’s never negative. So, going back to my original problem, only x = 2 is a solution
because only 2 is positive. So, where did this incorrect x = -1 come from? Well, suppose the
original equation had been this: -x = √ (x + 2). If you solve this equation by squaring both
sides, then you get the same quadratic equation as before, with the same solutions, 2 and -1.
Now, x = -1 is a solution of this new equation, but x = 2 isn’t. Try it and see.
Alan Borthwick
So, remember that the solutions of a squared equation may not be the solutions of the original
equation.
SallyAnn Keizer
Now I’m going to look at another problem. Look at this equation. It’s the sort of equation you
might come across when studying conics.
9x2 -16y2 = 144
The task is to make y the subject of the equation and solve the y. So, I have minus 16y2 =
144, bringing the 9x2 over to the other side, becomes -9x2, and multiplying through by -1, that
gives me 16y2 = 9x2, that now becomes a ‘+’, -144, and dividing through by 16, that gives me
y2 = (9x2 – 144)/16
But it’s not immediately obvious what to do next, and how to get the expression down to a
simple form to solve for y. So, I’m going to let Alan intervene here with some more rules for
how to handle square roots.
Alan Borthwick
Thanks, SallyAnn. Right, quiz time. Coming up is half a dozen identities, not all of which are
correct, so be warned, don’t believe everything you see. Now, as before, a and b are always
positive, and I’m looking for a list of correct relationships for how you might manipulate sums
and products of square roots. Oh, and by the way, you’ll see in a moment the significance of
this rubbish bin. Right, first of all, √(a + b) = √a + √b. Or is it?
SallyAnn Keizer
It doesn’t look quite right to me.
Alan Borthwick
Okay, well let’s have a look. Do a test. That’s always a good principle, by the way, in maths,
do a test. If a is 9, and b is 4, then we get √(9 + 4) = √9 + √4. √13 = 3 + 2? No way, into the
rubbish bin. Second one, √(a x b) = √a x √b?
SallyAnn Keizer
I’m more comfortable with that formula.
Alan Borthwick
Okay, let’s test. Letting a = 9 and b = 4, what happens? Let’s watch. So, we end up with the
√36, which is 6 = 3 x 2. That particular case is true. It doesn’t always mean, by the way, that
a general case would be true, but it is in this case. So, the general case is the first rule we’re
going to use here on our list. Right, let’s have the next one. The √(a – b), is that = √a - √b?
SallyAnn Keizer
That’s very similar to the first one, which was wrong, but let’s check it.
Alan Borthwick
Okay. Well, using 9 and 4 again, on the left, we have the √5, and is that = 3 - 2? I don’t think
so. Into the rubbish bin. Right, 4, √(a / b), is that = √a / √b? Let a =, as we said before, the
number 9 and b 4, let’s see what happens.
So, we’re getting on both sides, 3 / 2 = 3 / 2, correct in the particular case, so the general
case becomes our second rule. There’s another couple I’d like to mention, by the way, before
we finish. The first is that √a2 is always = a. And, secondly, (√a)2 = a. So, we have these
one, two, three, four very important identities to do with square roots.
SallyAnn Keizer
Right, well, with all of that on board, I can now go back to my last example, and manipulate it
in the right way to make y the subject of the equation. Now, I said before that
y2 = (9x2 – 144)/16
Now, if I take the square root of both sides, that gives me
y = + or - √((9x2 – 144)/16)
Now, using Alan’s second rule, that gives me
y = + or - √(9x2 – 144)/√16
Now, it might be tempting here just to square root the 9x2 and square root the 144, but if you
remember, that was one of the rules that Alan threw into the rubbish bin. But, what I can do
here is bring out the factor 9, so that gives me
y = + or - √, bringing out the factor 9, of 9(x2 – 16), because 16 times 9 equals that 144. All
over, not forgetting, √16, which is 4.
y = + or - √9(x2 – 16)/4
So, that gives me, using Alan’s first rule
y = + or – (√9 x √ (x2 – 16))/4
Nearly there. So,
y = + or - √9 is 3, so that’s 3/4 √ (x2 – 16). So,
y = + or – 3/4 √ (x2 – 16).
Alan Borthwick
Well, that might have looked difficult, but if you remember the rules for manipulating square
roots and make sure you’re careful, you should be fine. Good luck and goodbye.
SallyAnn Keizer
Goodbye.