We're asked to solve the given equation.
Notice how the given equation is a
degree two equation, which means the
equation is a quadratic equation.  To
solve the equation by factoring, the
first step is to set one side of the
equation equal to zero. Looking at the
right side of the equation, notice how
the degree two term is negative.
While it's not required it's typically
preferred to have the degree two term
positive when solving a quadratic equation
by factoring.
So let's set the right side of the
equation equal to zero by adding 2t
squared to both sides as well as
subtracting eleven t to both sides. So
beginning with the given equation,  for
the first step let's add two t squared to
both sides.
On the left side notice how these are
not like terms and therefore the sum is
two t squared plus five t. On the right side
negative two t squared plus two t squared is 0.
The right side simplifies to eleven t and
to set  the right side equal to 0
we'll subtract eleven t on both sides.
Simplifying
on the left we have two t squared and then five t minus eleven t is negative six t which will
write as minus six t. On the right side
eleven t  - eleven t is zero.  Now that we have the
right side equal to 0, we factor the left
side.
The first step in factoring is the
factor out the greatest common factor.
The greatest common factor between two t squared and six t is two t.  If we don't recognize this
we can write  negative two t squares as two times t times t and the prime factorization of 6
is 2 times 3.
So we can write six t as two times three times t. Looking at the factors of both terms, notice
how they both share a common
factor of two as well as the common
factor of t,  which indicates the greatest
common factor is two t.
So we factor out to two t from the
binomial on the left.  If we factor out to
two t from two t squared, we're left with one factor of t which we can see here.  And then - we we
factor out to two t from six t we're left with a factor of 3 which we see here.
So the factored form of the left side of
the equation is two t times the quantity t
minus three.  Now to solve we use the zero product property which indicates if this
product is equal to zero then either the
first factor of two t must equal 0 or
second factor of t minus 3 must equal 0.
Now we solve these two equations to
find our solutions.  To solve the equation
Two t equals zero, we divide both sides by
two. Simplifying we get t equals zero or to solve
t minus three equals zero we add three to both sides.
Simplifying we get t equals three.
so the given equation has two solutions
t equals zero for t equals three.
I hope you found this helpful.
