### Video Transcript

Given that negative 10 is a root of
the equation two π₯ squared plus 13π₯ minus 70 equals zero, what is the other
root?

Weβre told that negative 10 is a
root of our equation, which means that our quadratic must be equal to zero when π₯
is equal to negative 10. Essentially, itβs a solution to the
equation two π₯ squared plus 13π₯ minus 70 equals zero. Now, what this actually means is
that π₯ plus 10 must be a factor of two π₯ squared plus 13π₯ minus 70. Two π₯ squared plus 13π₯ minus 70
can therefore be written as π₯ plus 10 times some other binomial.

Now, letβs give that binomial a
form. Letβs say itβs in the form ππ₯
plus π, where π and π are real constants. What weβre going to do is
distribute the parentheses on the right-hand side of this equation and see what we
get. We begin by multiplying π₯ by
ππ₯. Thatβs ππ₯ squared. We then multiply the outer
terms. Thatβs π₯ times π, which is
ππ₯. Next, we multiply the inner
terms. Thatβs 10 times ππ₯, which is
10ππ₯. And finally, we multiply 10 by π,
to give us 10π. So we find that this is equal to
two π₯ squared plus 13π₯ minus 70.

And we now use a process called
comparing coefficients. We look at the coefficients of our
various terms. Letβs begin by comparing our
coefficients of π₯ squared. On the left-hand side, we have a
two. And on the right-hand side, the
coefficient of π₯ squared is π. So when we compare coefficients of
π₯ squared, we find π is equal to two.

We could next compare coefficients
of π₯. In fact though, weβre going to
compare constants. We might say these are the
coefficients of the π₯ to the power of zero terms. On the left-hand side, our constant
is negative 70. And on the right-hand side, we have
10π. So negative 70 equals 10π. And so weβre going to divide by 10
to solve for π. π is therefore equal to negative
seven. This means that our quadratic
expression can be written as π₯ plus 10 times two π₯ minus seven. Weβve replaced π and π with their
solutions.

But we know that weβre using this
to solve the equation two π₯ squared plus 13π₯ minus 70 equals zero. We already know that we have one
root of negative 10. Thatβs found by setting π₯ plus 10
equal to zero. Weβre now going to set two π₯ minus
seven equal to zero and solve for π₯. Weβll add seven to both sides of
this equation so that two π₯ is equal to seven. And then weβll divide through by
two. So π₯ is equal to seven over two or
3.5. And we could check this solution by
substituting π₯ equals seven over two into our original equation, making sure that
it is indeed equal to zero.