Video Transcript
If we take the square of Noah’s age
now, in years, and subtract five times his age two years ago, the result is 160. What is Noah’s age now, in
years?
In this question, we’ve been given
some information about Noah’s age, which includes the square of his current age. And this indicates that we might be
able to form and solve a quadratic equation to find out how old Noah is now. So let’s begin by calling Noah’s
current age 𝑥.
Our equation involves subtracting
five times Noah’s age two years ago from the square of Noah’s age now. So let’s write down what this means
in terms of 𝑥. We have 𝑥 squared, that’s the
square of his age now, minus five times 𝑥 minus two, which is his current age minus
two years. In other words, 𝑥 minus two is his
age two years ago. And we’re told that this is all
equal to 160.
So now if we distribute the
parentheses on the left-hand side, this gives 𝑥 squared minus five 𝑥 minus five
times negative two equals 160, which is 𝑥 squared minus five 𝑥 plus 10 equals
160. Now subtracting 160 from both
sides, we have 𝑥 squared minus five 𝑥 plus 10 minus 160 is equal to zero, which
leaves us with the quadratic equation 𝑥 squared minus five 𝑥 minus 150 equals
zero. Remember, we want to find Noah’s
age now, that’s 𝑥, so we need to solve this equation for 𝑥. We can actually do this by
inspection, so let’s make some space and see what we can come up with.
What we’re looking for is two
numbers whose product is negative 150 and whose sum is negative five. Now we know that 15 times 10 is
150. But we need one of our factors to
be negative since we require a product of negative 150. So now looking for an appropriate
sum, we know that negative 15 plus 10 is equal to negative five. Hence, applying the negative sign
to 15, we have negative 15 times 10 equals negative 150 and negative 15 plus 10
equals negative five.
We’re then able to factor our
equation into 𝑥 minus 15 times 𝑥 plus 10 equals zero. And this means either 𝑥 minus 15
equals zero or 𝑥 plus 10 equals zero. Adding 15 to both sides in the
first equation gives us 𝑥 equal to 15. And subtracting 10 from both sides
of the second equation gives us 𝑥 equal to negative 10. But since age can’t be negative,
the only valid solution is 𝑥 equals 15. Hence, if taking the square of
Noah’s age now and subtracting five times his age two years ago gives 160, Noah is
now 15 years old.
