To finish our problem, we need
to solve the quadratic equation
x squared minus
6x minus 27 equals
0, which we can do using
the quadratic formula, which
says that our roots-- the 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.
In this example, our coefficient
of x squared-- a-- is 1,
our coefficient of
x-- b-- is negative 6,
and our constant coefficient--
c-- is negative 27.
So putting this
into our formula,
we get that our roots
are negative negative 6
plus or minus the square
root of negative 6
squared minus 4 times 1 times
negative 27 all over 2 times 1.
And this can be simplified
because negative negative 6
is just six, negative 6 squared
is 36, negative 4 times 1 times
negative 27 is positive 108,
and 2 times 1 is just 2.
So our roots are 6 plus or
minus the square root of 36
plus 108 over 2.
And 36 plus 108 is 144
whose square root is 12,
so our roots are 6 plus
or minus 12 over 2.
6 plus 12 is 18, which we
have over 2, and 6 minus 12
is negative 6, which
we again have over 2.
Reducing these, we get that
our roots are 9 and negative 3.
So we can conclude
that the solutions
to x squared minus
6x minus 27 are
x equals 9 and x
equals negative 3.
