Video Transcript
Find the positive number which is
66 less than twice its square.
To answer this question, we’re
going to need to use some algebra, and we’re going to need to form an equation. So let’s introduce a letter to
represent this positive number. We can call it whatever we like;
let’s call it 𝑥. We’re told that 𝑥 is 66 less than
twice its square. Now, the square of 𝑥 can be
written as 𝑥 squared. Twice its square means we’re
multiplying this by two, so that gives two 𝑥 squared. And we then want 66 less than this
value. So we have the expression two 𝑥
squared minus 66. But remember, this is equal to the
number itself. So as an equation, we have two 𝑥
squared minus 66 is equal to 𝑥. We’ve therefore formulated this
problem as a quadratic equation. And we now need to solve it in
order to find the value of 𝑥.
We want to collect all of the terms
on the same side of the equation. Which we can do by subtracting 𝑥
from each side, giving two 𝑥 squared minus 𝑥 minus 66 equals zero, which is a
quadratic equation in its most easily recognizable form. Now, we want to solve this
quadratic equation by factoring. But notice that the coefficient of
𝑥 squared, the leading coefficient, is not equal to one. So we’re going to use the method of
factoring by grouping. We look for two numbers whose sum
is the coefficient of 𝑥, that’s negative one, and whose product is the product of
the coefficient of 𝑥 squared and the constant term. That’s two times negative 66 which
is negative 132.
The more familiar you are with your
times tables, the more quickly you’ll spot these two numbers. They’re negative 12 and positive
11. Now, the next step may look a
little strange. What we’re going to do is take that
term of negative 𝑥 and rewrite it using these two numbers. We’re going to rewrite it as
negative 12𝑥 plus 11𝑥. So we now have our quadratic
equation with four terms, two 𝑥 squared minus 12𝑥 plus 11𝑥 minus 66 is equal to
zero. Next, what we’re going to do is
split this quadratic equation in half. And we’re going to factor the two
halves separately.
Looking at the first half, two 𝑥
squared minus 12𝑥, these two terms have a common factor of two 𝑥. And if we take this out, we’re then
left with 𝑥 minus six. So the first half factors as two 𝑥
multiplied by 𝑥 minus six. Looking at the second half of our
quadratic, 11𝑥 minus 66, these two terms have a common factor of 11. And once we’ve factored by 11,
we’re left with 𝑥 minus six inside the parentheses. So the second half of our quadratic
factorizes as 11 multiplied by 𝑥 minus six.
Now, the key point, and this will
always be the case if a quadratic equation can be solved by factoring, is that the
two halves of our expression now have a common factor of 𝑥 minus six. We therefore factor the entire
quadratic by 𝑥 minus six. For the first term, we have to
multiply by two 𝑥. And for the second, we have to
multiply by positive 11. So our quadratic can be written as
𝑥 minus six multiplied by two 𝑥 plus 11. And it’s now in a fully factored
form. Remember, it will always be the
case that the two halves of your expression share a common factor if the quadratic
equation can be factored.
If you go through this method and
you find the two halves don’t share a common factor, then you’ve either made a
mistake or the quadratic equation you’re working with can’t actually be
factored. And you need to use another method
to solve it such as the quadratic formula or completing the square, if you’re aware
of these. Anyway, our quadratic equation can
be factored, and we have its factored form. So the next step is to take each
factor in turn and set them equal to zero.
We have 𝑥 minus six equals zero or
two 𝑥 plus 11 equals zero. To solve the first equation, we
need to add six to each side, giving 𝑥 equals six. And we can solve the second
equation in two steps. First, we subtract 11 from each
side, giving two 𝑥 equals negative 11. And then, we can divide by two,
giving 𝑥 equals negative 11 over two or negative 5.5. Now, both of these are valid
solutions to our quadratic equation. But if we look back at the
question, we were told that this number that we’re looking for must be positive. Which means it can’t be equal to
negative 11 over two. So we can eliminate this value. Our solution, then, is that 𝑥 is
equal to six. But let’s check this.
This number needs to be 66 less
than twice its square, so we have two times six squared minus 66. That’s two times 36 which is 72
minus 66 which is indeed equal to six, the number itself. So this confirms that our solution
is correct. So we have our answer to the
problem. The positive number we’re looking
for is six.
