Tom is solving the equation three
multiplied by 𝑥 plus six equals 12. Here is his working. Three multiplied by 𝑥 plus six
equals 12. Three 𝑥 plus six equals 12. Three 𝑥 equals six and 𝑥 is equal
to two. Part a) Explain what is wrong with
Katherine is solving the equation
two 𝑥 squared minus 13 equals 19. Here is her working. Two 𝑥 squared minus 13 equals
19. Two 𝑥 squared equals 32. 𝑥 squared equals 16. 𝑥 equals four. Part b) Explain what Katherine did
wrong in her working.
First of all, if we look at part a
and we look at Tom’s working, what we see here is actually this is where the mistake
has happened because we have three 𝑥 plus six. However, if we actually expand
bracket three multiplied by 𝑥 plus six, we’re actually gonna get a different
answer. So now, I’m actually gonna show you
how you get the answer and what it is.
So first of all, we’re gonna do
three multiplied by 𝑥 and this actually gives us three 𝑥. So great, this part is correct. However, what we should have done
was three multiplied by positive six, which would give this positive 18. So we actually end up with three 𝑥
plus 18, not plus six. So that would have been wrong.
And actually, this is one of the
most common mistakes made by students because what they do is they actually multiply
the first term and then great they multiply often the term that involves the
variables, so 𝑥 or 𝑦 or whatever, and they sometimes neglect the multiplication
when they have the number outside the bracket multiplied by possibly a number inside
the bracket or the second term. What they also might do is to
actually add them. So be careful of that when you do
this kind of question.
So therefore, we can say what is
wrong with Tom’s working is that the bracket hasn’t been expanded correctly. So we should have had three
multiplied by 𝑥 plus six is equal to three 𝑥 plus 18. So that’s part a solved. Let’s move on to part b.
So for part b, what we’re gonna do
is look at Katherine’s working. And she’s trying to solve the
equation two 𝑥 squared minus 13 equals 19. So what we’re gonna do with
Katherine’s solution is have a look at it a row at a time. So we can see that the first thing
that Katherine have done is actually added 13 to each side. And that gets us two 𝑥 squared is
equal to 32. So this is correct. So next, what she’s actually done
is divided both sides of the equation by two which gives 𝑥 squared is equal to
16. So again, this is correct, so no
mistakes so far. So what Katherine did next was she
actually took square root of each side of the equation. So we got 𝑥 is equal to four.
And on the face of it, this
actually looks right. What’s wrong with this? Well, actually, what’s wrong with
this is that we’ve only got half of the answer because what Katherine has done is
said that 𝑥 is equal to four. However, this only relates to
positive four. And actually, root 16 could be
equal to positive four or negative four. But why is this? Well, this is the case because if
we have negative four and we square it, we get an answer of 16. But also if we get four and square
it, we’ll also get an answer of 16.
What I draw your attention to at
this point is quick tip and that’s if you’re doing a negative four squared on your
calculator, then make sure you include brackets because if you don’t what the
calculator will often do is actually giving an answer with negative 16 because what
it thinks you’re doing is four squared and then making it negative. So make sure you include those
brackets so you don’t actually make a mistake in your calculation. And the reason we actually get
positive 16 when multiplying negative four by negative four is because if we
multiply a negative by a negative, we’ll get a positive. So that’s why we can actually have
So therefore, we can say that when
Katherine was solving the equation two 𝑥 squared minus 13 equals 19, the errors in
her actual working were that she gave the answer of 𝑥 is equal to four, which
should be 𝑥 is equal to negative or positive four. And that’s because two numbers give
16 when they are squared: four and negative four.