### Video Transcript

Find the values of 𝑘 for which the
quadratic equation 𝑘 plus four 𝑥 squared plus 𝑘 plus one 𝑥 plus one equals zero
has equal roots and then find those roots.

The key piece of information given
in the question is that the roots to the equation are equal. And in order to use this piece of
information, we can look at the quadratic formula which gives us solutions to
equations of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero. And these solutions are 𝑥 is equal
to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 over two
𝑎.

Now, if the roots or solutions to
the equation are equal, then this means that there’s only one unique solution. And if we look at the quadratic
formula, we can see that the only way for that to be one solution is if the square
root is equal to zero.

Now, the part in the square root of
the quadratic formula has another name. It’s called the discriminant. And so we can say that since our
equation has equal roots, its discriminant must, therefore, be equal to zero. Now, let’s find what the
discriminant of our equation will be.

If we compare our equation given in
the question to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero, we can see that 𝑎
is equal to 𝑘 plus four, 𝑏 is equal to 𝑘 plus one, and 𝑐 is equal to one. Next, we can set our discriminant
equal to zero. So that means 𝑏 squared minus four
𝑎𝑐 equals zero. And we simply substitute in our
values for 𝑎, 𝑏, and 𝑐. And we obtain that 𝑘 plus one all
squared minus four lots of 𝑘 plus four times one is equal to zero.

And now, we can expand the
brackets. And we obtain that 𝑘 squared plus
two 𝑘 plus one minus four 𝑘 minus 16 is equal to zero. And now, we can simplify this to 𝑘
squared minus two 𝑘 minus 15 is equal to zero.

And now, we can factorize this by
finding factor pairs of negative 15 which sum to negative two. And we found that the factor pair
of three and negative five sum to give negative two. And so we can factorize this
equation to get 𝑘 plus three multiplied by 𝑘 minus five is equal to zero.

This tells us that 𝑘 plus three
equals zero or 𝑘 minus five is equal to zero. If 𝑘 plus three is equal to zero,
then this gives us a solution of 𝑘 is equal to negative three. And if 𝑘 minus five is equal to
zero, this gives us our other solution of 𝑘 is equal to five. And so now, we found all values of
𝑘 for which the quadratic equation has equal roots.

Next, we just need to find the
roots to our equation. In order to do this, we can use the
quadratic formula again. However, we know that for these
values of 𝑘, the discriminant is equal to zero. And since this is the case, the
quadratic formula simply becomes 𝑥 is equal to negative 𝑏 over two 𝑎. Substituting in our values of 𝑎
and 𝑏, we obtain that 𝑥 is equal to negative 𝑘 plus one over two timesed by 𝑘
plus four.

Next, we just need to substitute in
our values of 𝑘 equals negative three and 𝑘 equals five. For 𝑘 equals negative three, we
get 𝑥 is equal to negative negative three plus one over two multiplied by negative
three plus four. And this simplifies to two over
two, which gives us 𝑥 is equal to one. Then, for the value of 𝑘 equals
five, we have that 𝑥 is equal to negative five plus one over two multiplied by five
plus four. And this simplifies to negative six
over 18, which gives 𝑥 is equal to negative one-third.

Now, we have found the roots to our
equation. We have that when 𝑘 is equal to
negative three, 𝑥 is equal to one. And when 𝑘 is equal to five, 𝑥 is
equal to negative one-third. And this completes our solution to
this question.