Video Transcript
Simplify two π₯ squared plus five
π₯ minus three over π₯ plus three.
First, we recall that this fraction
line means divide. And so when we simplify, weβre
actually saying how can we divide two π₯ squared plus five π₯ minus three by π₯ plus
three. Well, once our division problem is
written as a fraction, we look to factor where possible. Now, itβs not possible to factor
the expression on the denominator. But we can factor the
numerator. Letβs look to factor two π₯ squared
plus five π₯ minus three. There are a number of ways of doing
this. One way is called the AC
method. Itβs called the AC method because
given a quadratic equation of the form ππ₯ squared plus ππ₯ plus π, we begin by
multiplying the value of π and π. In our equation, π, which is the
coefficient of π₯ squared, is two and π is negative three. Two multiplied by negative three is
negative six.
Our next step, is just like when we
factor a quadratic equation where the coefficient of π₯ squared is one. We look for two numbers that
multiply to make negative six and add to make five. Well, six multiplied by negative
one is negative six. And six plus negative one is
five. And so weβre going to look to split
this middle term up into six π₯ and negative one π₯. We now write our quadratic as two
π₯ squared plus six π₯ minus one equals three. Now, weβve not done anything
mind-blowing here. Weβve just rewritten our original
expression. If we were to now simplify the
expression on the right-hand side, that would take us back to the expression on the
left.
The next step is to consider the
two pairs of terms. Weβre going to factor each
pair. We see that two π₯ squared and six
π₯ have a highest common factor or a greatest common factor of two π₯. And so two π₯ squared plus six π₯
can be written as two π₯ times π₯ plus three. Similarly, negative one π₯ minus
three have a common factor of negative one. So when we factor this expression,
we get negative one π₯ plus three.
Notice now that each term contains
a factor of π₯ plus three. And so we can factor by π₯ plus
three. Two π₯ times π₯ plus three divided
by π₯ plus three gives us two π₯. Then negative one times π₯ plus
three divided by π₯ plus three gives us negative one. And so we fully factored our
quadratic. Itβs π₯ plus three times two π₯
minus one. And this is great because we could
now rewrite our fraction. Weβve replaced the quadratic with
its factored form. And we see itβs equal to π₯ plus
three times two π₯ minus one all over π₯ plus three.
Now that itβs in this form, we can
simplify our fraction as we would in numerical fraction by dividing through by any
common factors. In this case, we can divide through
by π₯ plus three. When we do, we see that our
expression fully simplifies to two π₯ minus one over one or simply two π₯ minus
one. And so the answer to our question
and, in fact, the answer to two π₯ squared plus five π₯ minus three divided by π₯
plus three is two π₯ minus one. Now, we did use something called
the AC method to factor our quadratic expression. You may be used to using some other
method. And thatβs absolutely fine as long
as you do indeed end up with π₯ plus three times two π₯ minus one.
