hi everyone today we're going to talk about how to use the quadratic formula to solve
for the roots of a quadratic function to complete this problem we'll identify components of
our quadratic function plug them into the quadratic formula and then simplify the quadratic
formula to solve for the roots of the function let's take a look in this particular problem
we've been asked to use the quadratic formula to find the roots of the function f of x is
equal to one fourth x squared minus two x plus three the first thing we need to recognize
when we're talking about the quadratic formula is that we can only use the quadratic formula
when we have a quadratic function and in this case we do have a quadratic function because
our function is in the form a x squared plus bx plus c which is the form of a quadratic
function this format for a quadratic function recognizes that there are three constant coefficients
a is a constant coefficient and in our case that's one fourth which is the constant coefficient
on the front of x squared we have a constant coefficient of b which is in front of an x
to the first variable and in our case that's negative two it's always important to include
the negative sign that's there and then we have the constant value here c with no x variable
attached and ours in this case is just three so because we have the same format coefficient
multiplied by x squared plus or minus coefficient times x to the first power plus a constant
we know that we have a quadratic function so given that we have a quadratic function
with these three values we can use the quadratic formula to solve for the roots of this function
normally given a function like this our first choice to solve for the roots is just to factor
the function but that's not always possible and when we can't factor it one of the things
we can do is use the quadratic formula so as you can see with our quadratic formula
we have the same values shown in our quadratic formula we have b we have a here and we have
c here so let's go ahead and use the quadratic formula to solve for the roots of our function
so we'll say that x the root of our function will be negative b we know that b here is
negative two always include the negative sign if it is there so negative negative two plus
or minus the square root of b squared we know that b is negative two so we'll plug in negative
two minus four times a we know that a is one fourth times c we know that c is three so
we enter all those values that's all underneath our square root sign and then we divide the
entire numerator here by two times a and a and our quadratic function is one fourth so
once we have all of our values plugged in we just need to simplify as much as we can
so we have a negative negative two which will give us of course a positive two plus or minus
the square root negative two squared gives us four four and one fourth here will cancel
with one another and just become one and we'll be left with three so we'll get four minus
three underneath the square root sign all divided by two times one fourth which will
just give us one half when we simplify further we see that we'll get two plus or minus the
square root of one which we know will just be one all divided by one half dividing by
one half of course is the same thing as multiplying by two so we'll get two times two plus or
minus one so we'll get two roots here for x we'll get x equals two times two plus one
and we'll get x equals two times two minus one the first root of course will just give
us six the second root will give us two and that's it these are the roots of our function
that's how you use the quadratic formula to find the solutions to your quadratic function
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