Video Transcript
Find the solution set of the
equation five 𝑥 squared minus seven 𝑥 minus 32 equals zero, giving values to three
decimal places.
So, we have the quadratic equation
and we’re asked to find its solution set, which is just another way of saying solve
the equation. We’re told that we need to give
values to three decimal places. So, this is a big clue that the
quadratic equation we’ve been given won’t be able to be solved by factoring. And so, we need to use another
method. This is where we need to use the
quadratic formula, which we should make sure we learn off by heart. The quadratic formula tells us that
if we have the general quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals
zero. Then the solution set or roots of
this equation, if they exist, are given by 𝑥 equals negative 𝑏 plus or minus the
square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.
In order to apply this formula
then, what we need to do are determine the values of 𝑎, 𝑏, and 𝑐. 𝑎 is the coefficient of 𝑥
squared. So, for our equation, that’s
five. 𝑏 is the coefficient of 𝑥. So, for our equation, that’s
negative seven. And we must make sure that we
include the negative sign. 𝑐 is the constant term. In our equation, that’s negative
32. Now, all we need to do is
substitute these values of 𝑎, 𝑏, and 𝑐 into the quadratic formula. So, we have 𝑥 equals negative 𝑏 —
that’s negative negative seven — plus or minus the square root of 𝑏 squared, so
negative seven squared, minus four 𝑎𝑐 — that’s minus four times five times
negative 32. And this is all over two 𝑎. That’s two times five. You can break the formula down into
stages if you wish, or you can substitute straight into the formula. But you must be careful.
Let’s now look at simplifying. We have negative negative
seven. So that’s seven. And in the denominator, two times
five is 10. Let’s now look at the square
root. Firstly, we have negative seven
squared, and negative seven squared is equal to 49. If you’re using a calculator to
help with this, it’s really common to make a mistake when finding this value. Because what a lot of people do by
mistake is they type negative seven squared into their calculator, exactly as I’ve
written in orange here.
Now, when you do this, your
calculator will give negative 49, but that doesn’t mean your calculator has made a
mistake. It means you’ve made a mistake in
the question you’ve asked it. Negative seven squared means
negative one multiplied by seven squared. And if you recall BIDMAS or the
order of operations, we work our indices or powers before multiplication. So, negative seven squared means
negative one multiplied by seven squared, which is negative one multiplied by 49,
giving negative 49. What we should’ve typed into our
calculator was negative seven enclosed in brackets squared. So that ensures that our calculator
is taking a negative value first and then squaring it. This is a really common mistake,
and one you need to make sure you watch out for.
Continuing under the square root
then, we are subtracting four times five times negative 32. Now, four times five times negative
32 is negative 640. So we are subtracting negative 640,
which means overall we’re adding 640. But again, you need to be really
careful with the negative signs here. Simplifying under the square root,
49 plus 640 is 689. So, we have that 𝑥 is equal to
seven plus or minus the square root of 689 over 10.
Now, we’re asked to give our values
as decimals, so we now need to use our calculator to evaluate each of these. The first root or value in our
solution set is seven plus the square root of 689 all over 10, which is equal to
3.32488 continuing. The second root or second value in
our solution set is seven minus the square root of 689 all over 10. And that’s equal to negative
1.92488. We’re asked to give the values to
three decimal places, and in each case, the fourth decimal place is an eight. So, we round up. We can then give our two roots in
set notation. The solution set to this equation
is 3.325, negative 1.925.