Video Transcript
In this video, we will learn how to
solve quadratic equations with no linear term using the square root property. Let’s begin this video by recapping
what quadratic equations are.
A quadratic equation is an equation
of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are
constants and 𝑎 is not equal to zero. The reason that 𝑎 is not equal to
zero is if it was we wouldn’t have any term in 𝑥 squared and then the equation
would just be a linear equation. An example of a quadratic equation
could be two 𝑥 squared plus 𝑥 over two minus three equals zero.
When it comes to solving a
quadratic equation, that really means we’re trying to find the value of 𝑥. Quadratic equations will have up to
two different solutions for 𝑥. And there are a number of different
ways we can solve. For example, we could solve a
quadratic equation by factoring or even by drawing a graph. In this video, we’ll look at how we
can use the square root property. We can find the square root of a
quadratic equation when it has no linear term. That’s the term in 𝑥. So it would be of the form 𝑎𝑥
squared plus 𝑐 equals zero. Let’s look at an example.
Let’s say that we have the equation
𝑥 squared equals 25. To solve this for 𝑥, we could take
the square root of both sides of this equation. And since the square root of 25 is
equal to five, we could say that 𝑥 is equal to five. However, we should also remember
that negative five multiplied by negative five also gives an answer of 25. So we need to consider both the
positive and negative roots. We can do this by using the plus or
minus symbol. So 𝑥 equals plus or minus five
indicates two different solutions: 𝑥 equals positive five and 𝑥 equals negative
five. But the plus or minus symbol allows
us to write this more conveniently. Let’s now have a look at some
questions.
Solve 𝑥 plus six squared equals
four.
We can recall that when we’re
solving an equation, we’re looking to find the value of 𝑥. So let’s see how we would set about
solving this equation. One method of solving this equation
would be by taking the square root of both sides. This gives us 𝑥 plus six equals
the square root of four. But we should remember that when
we’re taking square roots, we need to consider both the positive and negative
roots. And we can do this by using the
plus or minus symbol. This is equivalent to saying 𝑥
plus six equals positive square root four and 𝑥 plus six equals negative square
root four. Taking the square root of four
gives us an answer of two. So on the right-hand side we have
plus or minus two.
In order to find 𝑥 by itself, we
subtract six from both sides of the equation. We can now write the two solutions:
𝑥 equals positive two subtract six or 𝑥 equals negative two subtract six. Positive two subtract six gives us
negative four, and negative two subtract six gives us negative eight. So we have 𝑥 equals negative four
or 𝑥 equals negative eight. And this would be our answer to the
question. But it’s important to remember that
we’re not saying just one of these solutions would work. We’re saying that both would
work.
We can check our answers by
substituting both of these back into the original equation. When 𝑥 is equal to negative four,
negative four plus six would give us two. And two squared does equal
four. And when 𝑥 is equal to negative
eight, we’d have negative eight plus six, which is negative two. And negative two squared also
equals four. And there we have our answer.
But it’s worth noting that there is
at least one other method we could have used to solve this. We could have begun by expanding
the parentheses. 𝑥 plus six all squared is
equivalent to 𝑥 plus six multiplied by 𝑥 plus six. Using a method such as FOIL would
then have given us 𝑥 squared plus six 𝑥 plus six 𝑥 plus 36 equals four. And we could then simplify by
collecting our terms in 𝑥. In order to get all our terms on
one side and zero on the other side, we would then need to subtract four.
In order to solve this equation,
we’ll now need to factor. And so 𝑥 squared plus 12𝑥 plus 32
on the left-hand side factors into parentheses 𝑥 plus four and 𝑥 plus eight. To solve this, we have 𝑥 plus four
equals zero or 𝑥 plus eight equals zero, which means that 𝑥 is equal to negative
four or 𝑥 is equal to negative eight. This is the same answer that we got
when we found the square root. Both of these methods are valid
methods to solve this equation but there’s certainly a lot less working out involved
in finding the square root.
Let’s take a look at another
question.
Find the solution set of the
equation four over 𝑥 equals 𝑥 over nine.
In order to begin solving this, we
can take the cross product. Four multiplied by nine gives us
36, and 𝑥 multiplied by 𝑥 gives us 𝑥 squared. We can write this if we prefer as
𝑥 squared equals 36. We might now notice more easily
that this is a quadratic equation as we have a term in 𝑥 squared and no other
higher power of 𝑥.
The quickest way to solve this to
find the value of 𝑥 is to take the square root of both sides of the equation. When we’re taking square roots,
it’s important that we consider both the positive and negative solutions. So 𝑥 is equal to plus or minus the
square root of 36. The square root of 36 is six. So we have 𝑥 is equal to plus or
minus six. But we need to give our answer as a
solution set. Therefore, we give our answer that
it’s the set six and negative six.
Of course, it’s always worth
checking our answer by plugging our values back into the original equation. When 𝑥 is equal to six, we’d have
four over six equals six over nine. And we know that both of these
fractions are equivalent to two-thirds. When 𝑥 is equal to negative six,
we’d have negative four-sixths is equal to negative six-ninths. And both of these are equivalent to
negative two-thirds. So we’ve confirmed that the answer
is the set six and negative six.
Let’s have a look at another
question. If you wish, you can pause the
video and see if you can find the solution.
Find the solution set of the
equation 𝑥 minus five squared equals 100 in the set of rational numbers.
One of the methods we can use to
solve to find the value of 𝑥 is by rearranging this equation. We would begin by noticing that
because we have a square on the left-hand side, we’ll want to do the inverse
operation which is finding the square root of both sides of this equation. We’ll also want to consider the
positive and negative solutions to our square root. So we write that 𝑥 minus five is
equal to plus or minus the square root of 100. Remember that our equation that we
have is a quadratic equation and quadratic equations can have up to two different
solutions. We should recognize that 100 is a
perfect square. So this will give us plus or minus
10 on the right-hand side.
In order to find 𝑥 by itself, we
then must add five to both sides of this equation, giving us 𝑥 equals plus or minus
10 plus five. It’s a little bit neater to write
this as 𝑥 equals five plus or minus 10, but it doesn’t really matter. Now we need to find our two
different solutions. We’ll have 𝑥 equals five plus 10
and 𝑥 equals five minus 10. Five plus 10 is 15, and five minus
10 is negative five. And we indicate these two solutions
by saying 𝑥 equals 15 or 𝑥 equals negative five.
Let’s check these values in our
original equation. When 𝑥 is equal to 15, we’d have
15 minus five, which is 10, and 10 squared does give us 100. The second solution, when 𝑥 is
equal to negative five, we’d have negative five subtract five, which would be
negative 10. And negative 10 squared also gives
us 100. Finally, we need to give our answer
as a solution set. So we write the set negative five,
15.
In the next question, we’ll see how
sometimes we need to do a bit more rearranging before we find the square roots.
Determine the solution set of the
equation negative two 𝑥 squared plus 15 equals 𝑥 squared minus 12.
In order to begin solving this
quadratic equation, we need to collect together the like terms. We can begin by subtracting 𝑥
squared from both sides of the equation. This gives us negative three 𝑥
squared plus 15 equals negative 12. Next, we subtract 15 from both
sides, which gives us negative three 𝑥 squared equals negative 27. In the next step, we divide both
sides of this equation by negative three. So 𝑥 squared is equal to nine.
In the final step, we need to do
the inverse operation to squaring, which is taking the square root. However, we must remember both the
positive and negative square roots. So 𝑥 is equal to plus or minus
three. And we write this because both
positive three multiplied by positive three gives us nine and negative three
multiplied by negative three gives us nine.
It’s important that we remember
this plus or minus symbol when we’re solving quadratic equations as it will indicate
two different solutions to the quadratic equation. In this case, they’ll be 𝑥 is
equal to three or 𝑥 is equal to negative three. Here we give the answer in set
notation as negative three, three.
In the final question, we’ll see an
example where the square root method doesn’t give any solutions.
Determine the solution set of the
equation 44𝑥 squared plus nine equals zero, given that 𝑥 is an element of the set
of real numbers.
Let’s begin solving this equation
by rearranging it so that we isolate the 𝑥 squared term. So subtracting nine from both sides
of this equation gives us 44𝑥 squared equals negative nine. Next, we divide both sides by 44,
so 𝑥 squared is equal to negative nine over 44. We can then take the square root of
both sides of this equation, and we consider both the positive and negative
roots. We can therefore write that 𝑥 is
equal to plus or minus the square root of negative nine over 44.
However, you might notice that we
have a problem. And it’s the fact that we’re trying
to take the square root of a negative number. Neither of these solutions to 𝑥
gives us an answer that’s a real number as both of these take the square root of a
negative value. There are no solutions to this
answer in the set of real numbers. As a set then, the answer would be
the null set.
Before we finish with this
question, there’s a few things to note. Firstly, it doesn’t matter which
method we use to solve this. We wouldn’t have got any solution
in the real numbers. Secondly, if we were to consider
the graph of the function 𝑓 of 𝑥 equals 44𝑥 squared plus nine, having no
solutions for 𝑥 simply means that the graph doesn’t pass through the 𝑥-axis.
We’ll now summarize the key points
of this video. We saw how we can use the square
root method to solve a quadratic equation, and we can do this using the steps
below. Firstly, if it’s not already done,
we need to collect the 𝑥 squared terms together and collect the constant terms on
the other side of the equation. Then, we take the square root of
both sides of the equation. Finally, we must consider both the
positive and negative values of the roots. The plus or minus sign is a useful
way to indicate these two values.
