Welcome to a lesson on Vieta's Formula
for quadratic equations.
Vieta's formula relates
the coefficients of a
polynomial to the sum and product
of its roots or solutions.
For a quadratic equation,
if alpha and beta
are the roots of ax squared
plus bx plus c equals zero.
Then the sum S of the
roots is alpha plus beta
which equals negative b divided by a.
And the product P of the
roots is alpha times beta
which equals c divided by a.
And the quadratic equation
with roots alpha plus beta
can be written as x squared minus S
times x plus P equals zero.
Let's look at an example.
We're asked to find a quadratic equation
with roots 1/2 and negative
three using Vieta's formula.
Let's go ahead and let alpha equal 1/2
and beta equal negative three.
Let's first find the sum S,
which is equal to 1/2 plus negative three.
Let's write negative three as a fraction
with the denominator of one.
To determine the sum we need
the least common denominator
which is two.
We need to write negative three over one
as an equivalent fraction
with a denominator of two,
and multiplying the numerator
and denominator by two.
Notice now we have a
common denominator of two.
And the numerator is
one plus negative six,
which is negative five.
So now we know the sum
S equals negative 5/2.
Now let's determine the product P.
The product P is equal
to alpha times beta,
which is 1/2 times negative three,
or times negative three over one,
which gives us negative 3/2.
And now let's find an equation using
x squared minus S times
x plus P equal zero.
This gives us x squared
minus S is negative 5/2.
So we have negative 5/2 times x.
And then plus P is plus negative 3/2.
And this is equal to zero.
Let's go ahead and simplify this equation.
Remember subtracting a
negative is equivalent
to adding a positive.
So this gives us x squared plus 5/2x
and then adding a negative
is the equivalence
to subtracting a positive.
Let's write this as minus 3/2 equals zero.
So this quadratic equation does have roots
of 1/2 and negative three.
But let's find out an equivalent equation
that doesn't contain fractions.
We can clear the fractions
from the equation
by multiplying both sides by two.
On the left side we have 2x squared
plus two times 5/2 is five,
so we have plus 5x minus
two times 3/2 is three
giving us minus three equals zero.
This equivalent quadratic
equation also has roots
of 1/2 and negative three.
So it is important to recognize
there are an infinite number
of quadratic equations
that have these two roots.
We can create as many equations as we want
by multiplying both sides of
the equation by a constant.
But I also want to show how
we are going to determine
a quadratic equation using
the fact that the sum
is equal to a negative b divided by a.
And the product is c divided by a.
So we know the sum S is equal
to negative five divided
by two or negative 5/2.
But it's also equal to
negative b divided by a.
And we know that product P, which is equal
to negative 3/2 or negative
three divided by two
is equal to c divided by a.
We can use these equations to determine
the values of a, b and c
and then write the equation using the form
ax squared plus bx plus c equals zero.
Notice both the denominators are a,
and both the denominators
of the fractions are two.
This tells us that a is equal to two.
Looking at this equation,
notice negative b is
equal to negative five,
which means b equals positive five.
And finally, notice here c
is equal to negative three.
So now using this form
of a quadratic equation,
we now substitute a, b and c.
This gives us 2x squared plus 5x
minus three equals zero.
The same equation we
found here on the left.
I hope you found this helpful.
