Completing
the Square: Quadratic Examples &
Deriving the Quadratic Formula (page
2 of 2)

Solve
x^{2} +
6x + 10 = 0.

Apply
the same procedure as on the previous page:

This
is the original equation.

x^{2}
+ 6x + 10 = 0

Move
the loose number over to the other side.

x^{2}
+ 6x = – 10

Take
half of the coefficient on the x-term
(that is, divide it by two, and keeping the sign), and square
it. Add this squares value to both sides of the equation.

Convert
the left-hand side to squared form. Simplify the right-hand
side.

Note: If you
don't know about complex
numbers
yet, then you have to stop at this step, because a square can't
equal a negative number! Otherwise, proceed...

Square-root
both sides. Remember to put the "±"
on the right-hand side.

Solve
for "x
=", and simplify as necessary.

x
= –3 ± i

If
you don't yet know about complex numbers (the numbers with "i"
in them), then you would say that the above quadratic has "no solution".
If you do know about complexes, then you would say that this quadratic
has "no real solution" or that is has a "complex solution".

Since
solving "(quadratic) = 0" for x
is the same as finding the x-intercepts
(assuming the solutions are real numbers), it stands to reason that this
quadratic should not intersect the x-axis
(since x-intercepts
are "real" numbers). As you can see below, the graph does not
in fact cross the x-axis.

This
relationship is always true. If you come up with a real value on the right-hand
side of the equation (a zero value is real, by the way; the square root
of zero is just zero), then the quadratic will have two x-intercepts
(or only one, if you get plus/minus of zero on the right side); if you
get a negative on the right-hand side, then the quadratic will not cross
the x-axis.

I'll
do one last "example". It has become somewhat
fashionable to have students derive the Quadratic
Formula themselves;
this is done by completing the square for the generic quadratic equation
ax^{2}
+ bx + c
= 0. While I can understand
the impulse (showing students how the Formula was invented, and thereby
giving an example of the usefulness of symbolic manipulation), the computations
involved are often a bit beyond the average student at this point. Here
is what the instructor is looking for:

Derive
the Quadratic Formula by solving ax^{2}
+ bx + c = 0.

This
is the original equation.

ax^{2}
+ bx + c = 0

Move
the loose number to the other side.

ax^{2}
+ bx = –c

Divide
through by whatever is multiplied on the squared term. Take
half of the x-term,
and square it.

Add the squared
term to both sides.

Simplify
on the right-hand side; in this case, simplify by converting
to a common denominator.

Convert
the left-hand side to square form (and do a bit more simplifying
on the right).

Square-root
both sides, remembering to put the "±"
on the right.

Solve
for "x
=", and simplify as necessary.

Whether
you're working symbolically (as in the last example) or numerically (which
is the norm), the key to solving by completing the square is to practice,
practice, practice. By so doing, the process will become a bit more "automatic",
and you'll remember the steps when you're taking the test.