In this example, we'll
find the solutions
to the quadratic equation x
squared plus 4x plus 1 equals
0 using the quadratic
formula, which
says that the solutions to ax
squared plus bx plus c equals
0 are given by the
formula negative
b plus or minus the
square root of b squared
minus 4ac all over 2a,
where our a, b, and c here
are 1, 4, and 1.
Plugging these into
our formula, we
get negative 4 plus or
minus the square root of 4
squared minus 4 times 1
times 1, all over 2 times 1.
We can simplify this, because 4
squared is 16, 4 times 1 times
1 is 4, and 2 times 1 is 2.
So we get negative 4 plus
or minus the square root
of 16 minus 4 all over 2.
16 minus 4 is 12, so we have
negative 4 plus or minus
the square root of 12 over 2.
So we found that the
solutions to x squared
plus 4x plus 1
equals 0 are given
by x equals negative 4 plus or
minus the square root of 12,
all over 2.
It's worth pointing
out, though, that we
can make this a bit nicer.
Because the square
root of 12 we can
write as the square
root of 4 times 3,
which we can split up as
the square root of 4 times
the square root of 3, which is
2 times the square root of 3.
So we can rewrite our
solutions as r1 and r2
are negative 4 plus
or minus 2 times
the square root of 3 over 2.
And dividing by 2,
this is negative 2
plus or minus the
square root of 3.
So the solutions to x squared
plus 4x plus 1 equals 0
are negative 2 plus the square
root of 3 and negative 2
minus the square root of 3.
It's worth remembering that
this was our first example
that we realized we couldn't
solve just by using factoring.
And we were able
to do this before
without the quadratic formula,
but it took us a lot longer.
This is the power of
the quadratic formula.
It allows us to solve difficult
equations much quicker.
