Video Transcript
The sum of the squares of two
positive real numbers is 542. Given that one of them is 18, find
the value of the other one.
So in this question, we’re not
looking to take a trial-and-error approach. We’re going to answer this question
by forming an equation. Let’s look at the information we’ve
been given. We’re told that the sum of the
squares of two positive real numbers is 542. We need to introduce letters to
represent these numbers. So at this point you may think,
let’s call one number 𝑥 and the other 𝑦. And if the sum of their squares is
542, then we can express this as an equation, 𝑥 squared plus 𝑦 squared equals
542. However, if we read on a bit in the
question, we see that we’re told that one of the numbers is 18. So we don’t actually need to use
two letters at all. We can use one letter, 𝑥, to
represent the unknown number and then our second number is 18. So we have the equation 𝑥 squared
plus 18 squared equals 542.
Now, what we’ve done here is form
an equation. And it is a quadratic equation
because the highest power of our variable, that’s 𝑥, that appears is two. Let’s now see how we go about
solving this equation. Firstly, on the left-hand side, we
can evaluate 18 squared; it’s equal to 324. We want to get 𝑥 on its own on the
left-hand side. So the next step is to subtract 324
from each side. When we do, we’re just left with 𝑥
squared on the left-hand side. And on the right-hand side, 542
minus 324 is 218.
Now, we have 𝑥 squared equals
218. And in order to work out the value
of 𝑥, we need to apply the inverse or opposite operation of squaring, which is
square rooting. So we take the square root of each
side of the equation. The square root of 𝑥 squared is
𝑥. And on the right-hand side, we take
the square root of 218. But it’s important that when we
solve an equation by square rooting, we remember to take plus or minus the square
root. We have then that 𝑥 is equal to
plus or minus the square root of 218.
Now, this is certainly the correct
solution to the quadratic equation we’ve formed. But if we look back at the
question, we were told that both of our numbers are positive real numbers. Which means that in terms of the
answer to the question, 𝑥 must take only a positive value. So whilst negative the square root
of 218 is a valid solution to the quadratic equation, it isn’t a valid solution to
this problem. So our only answer is 𝑥 equals the
square root of 218. Now, this surd can’t be simplified
any further, as 218 has no square factors other than one. And it makes sense to give an exact
answer, so leaving our answer in terms of a square root rather than a rounded
decimal. Our answer then is that the second
number is the square root of 218.
We can, of course, check this by
substituting our value of 𝑥 back into our equation or the information given in the
question and confirming that we do indeed get 542 for the sum of the squares of the
two numbers.
