### Video Transcript

Solve π₯ squared plus eight π₯ plus
11 is equal to zero. Write your answer in the form π
plus or minus root π, where π and π are integers.

Usually, we try to solve a
quadratic equation by factorizing first. However, the question has shown us
that our answer will be in surd form. This means it will need to be
solved either using the quadratic formula or by completing the square. Now, we canβt use either
method. However, completing the square does
involve fewer steps and therefore there are fewer opportunities to make a
mistake.

Letβs look at the process for
completing the square. A quadratic equation in completing
the square form would look like this: π₯ plus π all squared plus π. Now, π and π donβt necessarily
need to be integers and itβs important to note that they are not the same letters as
the ones in the question. We have used π and π though
because these are the letters that are most commonly used when describing completed
square form.

First, letβs find the value of π
which is the number inside the bracket. We do this by halving the
coefficient of π₯. In this equation, the coefficient
of π₯ is eight. So halving it gives us four. That gives us π₯ plus four all
squared as the first part of our completed square form. Now, letβs look at why we actually
do this.

Imagine we were going to expand π₯
plus four all squared back out again. Remember to square something
actually means to multiply it by itself. So π₯ plus four squared is the same
as π₯ plus four multiplied by π₯ plus four. We can use the FOIL method to
multiply two brackets. π₯ multiplied by π₯ is π₯ squared,
π₯ multiplied by four is four π₯, and four multiplied by π₯ is also four π₯. These first three terms do actually
give us π₯ squared plus eight π₯ which is what we wanted from our question. However, when we multiply the final
part of each bracket, thatβs four multiplied by four, we get 16 not 11.

To counteract this then, we
subtract 16 from our bracketed expression. Then, we replace π₯ squared plus
eight π₯ with π₯ plus four all squared minus 16 in our original equation. Negative 16 plus 11 is negative
five. So our completed square form is π₯
plus four squared minus five.

The question is asking us though to
solve π₯ squared plus eight π₯ plus 11 is equal to zero. We can now replace π₯ squared plus
eight π₯ plus 11 with our completed square form. Donβt be tempted to reexpand this
bracket out. Instead, weβll solve this equation
by applying the relevant inverse operations.

We can start by adding five to both
sides. That gives us π₯ plus four squared
is equal to five. Next, we can find the square root
of both sides. Remember a square root will always
have two solutions: a positive and a negative. So we can add this little symbol
which means plus and minus to show that weβve got two separate answers. Finally, weβll subtract four from
both sides. π₯ is, therefore, equal to negative
four plus or minus the square root of five.

We can compare this to the form
given in the question which was π plus or minus the square root of π. We havenβt been asked to identify
what π and π are. But we can see by comparing them to
our equation that π is negative four and π is five.