### Video Transcript

In this video, we will learn how to
evaluate and write a quadratic function in different forms.

We recall, firstly, that a
quadratic expression is one in which the highest power of the variable that appears
is two. So, the expression includes a
squared term and no terms with higher powers, such as cubed or to the power of
four. For example, the expression π₯
squared plus two π₯ minus three is a quadratic expression, as is two π¦ squared
minus three. The expression π minus one all
squared is also a quadratic expression. Because once weβve distributed the
parentheses, we see that this is equivalent to π squared minus two π plus one, and
so the expression includes an π squared term.

There are a number of different key
forms in which a quadratic function can be written. The first of these is the general
or expanded form. π of π₯ equals ππ₯ squared plus
ππ₯ plus π, where π, π, and π are constants and π must not be equal to
zero. Itβs usual to write the terms in
order of descending powers of the variable. So, we have the π₯ squared term
first, then the π₯ term if there is one, and finally the constant term if there is
one. Although this isnβt essential.

The second form is completed square
or vertex form. π of π₯ equals π multiplied by π₯
plus π all squared plus π, where π, π, and π are all constants and, again, π
must be nonzero. The form we use depends on which
features of the quadratic function or its graph weβre particularly interested
in. In this video, weβll mostly be
focusing on the completed square form because itβs useful for determining the
coordinates of the vertex and the equation of the axis of symmetry of a quadratic
graph.

The final form in which we can
express a quadratic function is its factored or factorized form. π of π₯ equals ππ₯ plus π
multiplied by ππ₯ plus π, where π, π, π, and π are constants. Although, it isnβt possible to
express every quadratic in this form. Letβs now consider some
examples. In our first example, weβll see how
we can use the completed square or vertex form of a quadratic function to determine
the coordinates of the vertex of its graph.

Find the vertex of the graph of π¦
equals five multiplied by π₯ plus one all squared plus six.

Here, we have a quadratic
function. And itβs been expressed in its
completed square or vertex form. In general, this form is π
multiplied by π₯ plus π all squared plus π. So, comparing the general form with
our quadratic, we see that the value of π is five, the value of π is one, and the
value of π is six. We were asked to use this form to
find the vertex or turning point of the graph of this quadratic function.

Now, we should recall that all
quadratic functions have the same general shape, which is called a parabola. When the value of π is positive,
the parabola will curve upwards. And when the value of π is
negative, the parabola will curve downwards. The vertex of the graph is its
minimum point in the first instance and its maximum point in the second. Our value of π in this question is
five, which is positive. So, weβre in this first instance
here. Weβre looking for the coordinates
of the minimum point of this graph. Letβs consider then how we can
minimize this quadratic function and for what value of π₯ this minimum will
occur. And then, weβll see how we can
generalize.

Letβs consider each part of the
function in turn, starting with π₯ plus one squared. Well, this is a square. And we know that squares are always
nonnegative. Whatever value of π₯ we choose,
once we have added one and then squared it, the result will be zero or above. So, the minimum value of π₯ plus
one all squared is zero. We then multiply this by five. But, of course, anything multiplied
by zero is still zero. So, the minimum value of five
multiplied by π₯ plus one all squared is still zero. We then add six, so the entire
function increases by six units, which means that its minimum value overall is
six. This gives the minimum value of the
function or the minimum value of π¦. So, it is the π¦-coordinate of the
vertex.

We then need to determine what
value of π₯ causes this to be the case. Well, itβs the value of π₯ such
that π₯ plus one all squared takes its minimum value of zero. Itβs, therefore, the value of π₯
such that the expression inside the parentheses, thatβs π₯ plus one, is equal to
zero. The solution to this simple linear
equation is π₯ equals negative one. So, we find that the π₯-coordinate
of the vertex is negative one. The coordinates of the vertex of
this graph then are negative one, six.

Now, letβs consider how we can
generalize. The π₯-coordinate of the vertex was
the value that made the expression inside the parentheses zero. In the case of the general
expression π₯ plus π, this will be the value negative π because negative π plus
π give zero. So, this is the π₯-coordinate of
vertex. The π¦-coordinate of six was the
value that was added on to this squared expression. So, in the general form, that will
be the value π. So, we can generalize. If a quadratic function is in its
completed square or vertex form π¦ equals π multiplied by π₯ plus π all squared
plus π, then the coordinates of its vertex are negative π, π.

Now, the quadratic function in this
question was given to us in its completed square form. In our next example, weβll remind
ourselves how to take a quadratic in its general form and write it in its completed
square form.

In completing the square for the
quadratic function π of π₯ equals π₯ squared plus 14π₯ plus 46, you arrive at the
expression π₯ minus π all squared plus π. What is the value of π?

Here, weβre given a quadratic in
two forms, its general or expanded form and its completed square form. These two expressions describe the
same quadratic, and therefore theyβre equal. We can, therefore, form an equation
π₯ squared plus 14π₯ plus 46 is equal to π₯ minus π all squared plus π.

Now, there are two approaches that
we can take to answering this question. The two approaches involve working
in opposite directions. In our first approach, weβll
manipulate the expression on the right-hand side of our equation to bring it into
its expanded form. Using whatever method weβre most
comfortable for squaring a binomial, we see that π₯ minus π all squared is
equivalent to π₯ squared minus two ππ₯ plus π squared. So, the expression on the
right-hand side becomes π₯ squared minus two ππ₯ plus π squared plus π.

As these expressions are equivalent
for all values of π₯, we can now compare coefficients on the two sides of the
equation. The coefficients of π₯ squared on
each side are one. And then, if we compare the
coefficients of π₯, we have 14π₯ on the left-hand side and negative two ππ₯ on the
right-hand side, giving the equation 14 equals negative two π. We can solve this equation for π
by dividing each side by negative two, giving π equals negative seven.

Now, we have actually completed the
problem because all we weβre asked for was the value of π. But suppose weβd also been asked to
determine the value of π. We could do this by comparing the
constant terms on the two sides of the equation. On the left, we have positive
46. And on the right, we have π
squared plus π. So, that gives us a second
equation; π squared plus π is equal to 46. We know that π is equal to
negative seven and negative seven squared is 49. So, substituting this value into
our equation, we have 49 plus π equals 46. And subtracting 49 from each side,
we find that π is equal to negative three.

This means that the completed
square or vertex form of our quadratic, using π equals negative seven and π equals
negative three, is π₯ minus negative seven all squared minus three. Of course, π₯ minus negative seven
is better written as π₯ plus seven. So, we can express this as π₯ plus
seven all squared minus three.

So, thatβs our first method in
which we expanded to the expression on the right-hand side. In our second method, weβll see how
we can work the other way. So, weβll start on the left-hand
side and bring it into its completed square form. We already know what weβre working
towards. Itβs π₯ plus seven all squared
minus three. Now, notice that the value inside
the parentheses of positive seven is exactly half the coefficient of π₯ in our
original equation. And this is no coincidence. This will always be the case. So, we begin by writing π₯ squared
plus 14π₯ plus 46 as π₯ plus seven all squared. Weβve halved the coefficient of π₯
to give the value inside the parentheses.

The trouble is, though, π₯ plus
seven all squared isnβt just equivalent to π₯ squared plus 14π₯; itβs equivalent to
π₯ squared plus 14π₯ plus 49. So, weβve introduced an extra
49. We, therefore, need to subtract
this so that the expression we have is still equivalent to π₯ squared plus 14π₯. This new expression of π₯ plus
seven all squared minus 49 is, therefore, exactly equivalent to π₯ squared plus
14π₯. We also need to include the
positive 46 so that the two sides of the equation are the same.

Now, that value 49 is, of course,
the square of seven. So, what weβre doing is subtracting
the square of the value inside our parentheses. The final step is just to
simplify. We have negative 49 plus 46, which
is equivalent to negative three. And so, weβve found that this
quadratic, in its completed square form, is π₯ plus seven all squared minus three,
which is the same as we found using our first method.

Comparing this then to the given
form of π₯ minus π all squared plus π, we would see that positive seven is equal
to negative π. Dividing or, indeed, multiplying
both sides of this equation by negative one, we find that π is equal to negative
seven. Using two methods then, thatβs
working in both directions, weβve found that the value of π is negative seven.

Completing the square is a little
bit more complicated when the value of π, thatβs the coefficient of π₯ squared in
the general form, is not equal to one. So, letβs consider an example of
this.

Which of the following is the
vertex form of the function π of π₯ equals two π₯ squared plus 12π₯ plus 11. And weβre given five answer
options.

So, we have a quadratic function in
its general or expanded form, and weβre asked to find its vertex form. Thatβs π of π₯ equals π
multiplied by π₯ plus π all squared plus π, where π, π, and π are constants we
need to find. To answer to this question then, we
need to write our quadratic function in its completed square form.

Now, this is slightly trickier than
usual because the coefficient of π₯ squared in our function isnβt one. It is two. Now, we can deal with this by
factoring by this coefficient. You can either factor from all
three terms giving two multiplied by π₯ squared plus six π₯ plus 11 over two. Or we can simply factor this
coefficient from the first two terms, giving two multiplied by π₯ squared plus six
π₯ plus 11. And personally, I think the second
method is easier.

What weβre now going to do is
complete the square on the expression within the parentheses. Thatβs π₯ squared plus six π₯. We begin by halving the coefficient
of π₯ to give the number inside the parentheses. So, we have π₯ plus three all
squared. And then, we need to subtract the
square of this value. So, we have π₯ plus three all
squared minus nine. A quick check of redistributing
these parentheses confirms that π₯ plus three all squared minus nine is indeed equal
to π₯ squared plus six π₯.

Now, we need to be very careful
here. All of this expression of π₯
squared plus six π₯ was being multiplied by two. So, we need to put a large set of
brackets or parentheses around π₯ plus three all squared minus nine, multiply it by
two. And then, we still have the 11 that
we were adding on. So, we found that our function π
of π₯ is equivalent to two multiplied by π₯ plus three all squared minus nine plus
11.

Next, we need to distribute the
two. So, we have two multiplied by π₯
plus three all squared and then two multiplied by negative nine, which is negative
18, and then plus 11. Remember that 11 is not being
multiplied by two. Finally, we just simplify negative
18 plus 11 is negative seven. So, we have our quadratic in its
vertex form two multiplied by π₯ plus three all squared minus seven.

Looking carefully at the five
answer options we were given because theyβre all quite similar, we see that this is
answer option (a). Now, in this question, we just
worked through the completing-the-square process ourselves and then determined the
correct answer option. It would actually have been
possible to eliminate a couple of the options straightaway though. If we look at the vertex form of
the function, we see that within the parentheses we have just π₯ plus π all
squared. We could, therefore, have
eliminated options (b) and (c) as they are not the vertex form of any function
because inside the parentheses they each have two π₯.

We could also have eliminated
option (e) because we can see that there is no two involved in this option. And as the coefficient of π₯
squared in our original function was two, weβd need a factor of two outside the
parentheses as we have in answer options (a) and (d). The only difference between options
(a) and (d) is the sign inside the parentheses. We have positive three for option
(a) and negative three for option (d). We should remember, though, that
the sign here is always the same as the sign of the coefficient of π₯ in the
original function. So, in this case, itβs
positive. In any case, we have our answer
though. The vertex form of this function is
π of π₯ equals two multiplied by π₯ plus three all squared minus seven.

Quadratic functions can be used to
model practical situations such as the path followed by an object or a worded
problem. In our final example, weβll see how
we can form a quadratic function from a description.

Two siblings are three years apart
in age. Write an equation for π, the
product of their ages, in terms of π, the age of the youngest sibling.

So, weβre told in this question to
use the letter π to represent the age of the younger sibling. We now want to find an expression
in terms of π for the age of the older sibling. Well, weβre told that the siblings
are three years apart in age. So, the older sibling is three
years older than the younger. An expression for the age of the
older sibling is, therefore, π plus three. The product of their ages means we
need to multiply them together. So, we take our two expressions of
π and π plus three and multiply them.

This is a quadratic equation. Because if we were to distribute
the parentheses, we have π multiplied by π giving π squared and π multiplied by
three giving three π. Either of these two forms would be
acceptable for the answer, but weβll give our answer as π equals π multiplied by
π plus three.

Letβs now review some of the key
points from this video. Firstly, one of the key forms in
which a quadratic can be written is its general or expanded form. π of π₯ equals ππ₯ squared plus
ππ₯ plus π, where π, π, and π are constants and π must be nonzero. We can also express quadratics in
their completed square or vertex form. π of π₯ equals π multiplied by π₯
plus π all squared plus π, where π, π, and π are constants and, again, π must
be nonzero.

Weβve also seen that for a
quadratic function expressed in its completed square or vertex form, then its
vertex, which will be a minimum when the value of π is positive and a maximum when
the value of π is negative, is at the point with coordinates negative π, π. Through our examples, we saw how we
can convert between these two key forms either by following the
completing-the-square process or by distributing the parentheses and simplifying the
resulting expression. Although we didnβt need to use it
in this video, we also know that some quadratic functions can be written in a
factored or factorized form.