Suppose alpha and beta are the roots
of X squared minus seven
X plus five equals zero,
find alpha squared plus beta squared.
We will answer this question
using Vieta's formula
for quadratic equations, which states
that if alpha and beta are the roots
of A X squared plus B
X plus C equals zero,
then the sum of the
roots, alpha plus beta,
is equal to negative B divided by A,
and the product of the
roots, alpha times beta,
equals C divided by A.
Notice for our given
equation A is equal to one,
B is equal to negative seven,
and C is equal to five,
which means alpha plus beta
is equal to negative,
and then B divided by A,
which is negative seven divided by one,
which simplifies to positive seven.
So alpha plus beta is
equal to positive seven,
and alpha times beta is
equal to C divided by A,
which is five divided by one,
which simplifies to five,
which means alpha times
beta is equal to five.
Now I do want to mention,
when we're given a quadratic equation
with a leading coefficient of one
or if it's in this form here,
which is the form that we have,
we can determine the sum of the roots, S,
by taking the opposite
of the coefficient of X.
Notice how here the coefficient
of X is negative seven
and the sum of the
roots is positive seven,
and the product of the
roots P can be found
by using the constant term,
and notice how here the
constant term is five,
which does give us the product,
but this does only work
when the leading coefficient is one.
And now let's determine
the value of alpha squared
plus beta squared.
Notice how we cannot
evaluate the expression
in this form because we
don't have alpha squared
or beta squared.
So this one's a little bit tricky.
We're actually going
to consider the square
of alpha plus beta.
Well the square of alpha plus beta
is equal to two factors
of alpha plus beta.
Again, this might not
make a lot of sense now,
but it will in a moment,
and now we multiply.
We have four products.
One, two, three, and, four.
This is equal to alpha
times alpha is alpha squared
plus alpha times beta,
which is alpha beta,
plus beta times alpha,
which is another alpha beta.
So we have plus two alpha beta,
and then we have plus beta times beta,
which is plus beta squared.
Notice on the right side of this equation
we do have an alpha
squared plus beta squared,
but we also have this plus two alpha beta.
Let's subtract this on
both sides of the equation.
If we subtract two
alpha beta on both sides
we have alpha plus beta
times alpha plus beta minus two alpha beta
is equal to alpha squared
plus beta squared, and
now we can determine
the value of alpha
squared plus beta squared
by using the expression on the left.
We know alpha plus beta is equal to seven.
So here we have seven times seven
and then minus two times alpha times beta.
Alpha times beta is equal to five.
So we have times five,
and this is equal to
alpha squared plus beta squared.
Well seven times seven is 49
and two times five is 10.
So now we know that alpha squared
plus beta squared is equal to 39.
I hope you found this helpful.
