### Video Transcript

Solve the following quadratic
equation for 𝑥: four 𝑥 squared plus four 𝑏𝑥 minus 𝑎 squared minus 𝑏 squared is
equal to zero.

Now, as we don’t know the values of
𝑎 and 𝑏, our solution for the roots of this quadratic equation will be in terms of
𝑎 and 𝑏. We have a variety of methods that
we can use to solve a quadratic equation. We could try factorizing, we could
use the quadratic formula, or we could complete the square. Now, this quadratic equation
doesn’t look like it will easily factorize with simple algebraic expressions. So instead, we’ll try applying the
quadratic formula.

The quadratic formula tells us that
the roots of the quadratic equation, 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to
zero, are given by 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏
squared minus four 𝑎𝑐 all over two 𝑎. Now, we need to be a little bit
careful here because the letters 𝑎 and 𝑏 appear in the quadratic formula, but they
also appear in our quadratic equation. And they have different meanings in
the two.

For our quadratic equation, the
value of 𝑎, which is the coefficient of 𝑥 squared, is four. The value of 𝑏, which is the
coefficient of 𝑥, is four 𝑏. And the value of 𝑐, which is the
constant term, is negative 𝑎 squared minus 𝑏 squared. Now, we can actually simplify the
expression for 𝑐 by multiplying every term in the bracket by this negative sign or
negative one. And it gives 𝑏 squared minus 𝑎
squared.

So we can substitute our values of
𝑎, 𝑏, and 𝑐 into the quadratic formula. And it gives 𝑥 is equal to
negative four 𝑏 plus or minus the square root of four 𝑏 all squared minus four
multiplied by four multiplied by 𝑏 squared minus 𝑎 squared all over two multiplied
by four. We now just need to work through
simplifying this expression for the two roots of the quadratic equation,
carefully.

Let’s look at simplifying within
the square root first of all. The first term within the square
root is four 𝑏 all squared. Now this is equal to four squared
multiplied by 𝑏 squared which is 16𝑏 squared, not four 𝑏 squared. You must make sure you square the
four as well as the 𝑏. So watch out for this common
mistake here.

The second term within the square
root is four multiplied by four multiplied by 𝑏 squared minus 𝑎 squared. Well, four multiplied by four is
16. And then we need to multiply this
by each term inside the bracket. So it gives 16𝑏 squared minus 16𝑎
squared. So our expression for 𝑥 becomes
negative four 𝑏 plus or minus the square root of 16𝑏 squared minus 16𝑏 squared
minus 16𝑎 squared all over eight.

Now, within the square root we can
multiply both terms inside that bracket by the negative or negative one sign in
front of it. And it will give 16𝑏 squared minus
16𝑏 squared plus 16𝑎 squared. The two lots of 16𝑏 squared, one
of which is positive and one of which is negative, cancel each other out. So within the square root, we’re
just left with 16𝑎 squared. This means that our expression for
𝑥 simplifies to negative four 𝑏 plus or minus the square root of 16𝑎 squared all
over eight.

Now to find the square root of 16𝑎
squared, this is equal to the square root of 16 multiplied by the square root of 𝑎
squared. And the square root of 16 is
four. And the square root of 𝑎 squared
is 𝑎. So the square root of 16𝑎 squared
is just four 𝑎. Our expression for the roots,
therefore, becomes 𝑥 equals negative four 𝑏 plus or minus four 𝑎 all over
eight. And it’s starting to look a lot
more simple.

We can also cancel a common factor
of four from every term. So we’re left with negative one 𝑏
or negative 𝑏 plus or minus one 𝑎 or 𝑎 in the numerator and two in the
denominator. These roots are now fully
simplified. So the solution to the given
quadratic equation is 𝑥 is equal to negative 𝑏 plus 𝑎 all over two or negative 𝑏
minus 𝑎 all over two.