The
Quadratic Formula: Solutions and the
Discriminant (page
2 of 3)

Solve x(x
– 2) = 4.
Round your answer to two decimal places.

I not only cannot apply
the Quadratic Formula at this point, I cannot factor either.

I can not claim that
"x
= 4, x – 2 = 4",
because this is not how "solving
by factoring"
works. I must first rearrange the equation in the form "(quadratic)
= 0", whether I'm factoring or using the Quadratic Formula. The
first thing I have to do here is multiply through on the left-hand side,
and then I'll move the 4
over:

Then the answer is:
x
= –1.24, x = 3.24, rounded
to two places.

For reference, here's
what the graph looks like:

There is a connection between the solutions
from the Quadratic Formula and the graph of the parabola: you can tell
how many x-intercepts
you're going to have from the value inside the square root. The argument
of the square root, the expression b^{2}
– 4ac, is called the "discriminant"
because, by using its value, you can discriminate between (tell the
differences between) the various solution types.

Solve 9x^{2}
+ 12x + 4 = 0.

Using a
= 9, b = 12,
and c
= 4, the Quadratic
Formula gives:

Then the answer is x
= ^{–2}/_{3}

In the previous examples,
I had gotten two solutions because of the "plus-minus" part
of the Formula. In this case, though, the square root reduced to zero,
so the plus-minus didn't count for anything.

This solution is called
a "repeated" root, because x
is equal to
^{–2}/_{3},
but it's equal kind of twice:
^{–2}/_{3} + 0
and ^{–2}/_{3}
– 0. You can also see
this repetition better if you factor: 9x^{2}
+ 12x + 4 = (3x + 2)(3x + 2) = 0,
so x
= ^{–2}/_{3}
and x
= ^{–2}/_{3}.
Any time you get zero in the square root of the Quadratic Formula, you'll
only get one solution.

The square-root part of
the Quadratic Formula is called "the discriminant", I suppose
because you can use it to discriminate between whether the given quadratic
has two solutions, one solution, or no solutions.

This is what the
graph looks like:

The parabola only just
touches the x-axis
at x
= ^{–2}/_{3};
it doesn't actually cross. This is always true: if you have a root that
appears exactly twice, then the graph will "kiss" the axis there,
but not pass through.

Solve 3x^{2}
+ 4x + 2 = 0.

Since there are no factors
of (3)(2)
= 6 that add up to
4,
this quadratic does not factor. But the Quadratic Formula always works;
in this case, a
= 3, b = 4, and
c
= 2:

At this point, I have
a negative number inside the square root. If you haven't learned about
complex numbers
yet, then you would have to stop here, and the answer would be "no
solution"; if you do know about complex numbers, then you can continue
the calculations:

If you do not know
about complexes, then your answer would be "no solution". If
you do know about complexes, then you would say there there is
a "complex solution" and would give the answer (shown above)
with the " i
" in it. But whether or not you know about complexes, you know
that you cannot graph your answer, because you cannot graph the square
root of a negative number. There are no such values on the x-axis.
Since you can't find a graphable solution to the quadratic, then reasonably
there should not be any x-intercept
(because you can graph an x-intercept).

Here's the graph:

This relationship is always
true: If you get a negative value inside the square root, then there will
be no real number solution, and therefore no x-intercepts.
(The relationship between the value inside the square root, the type of
solutions, and the number of x-intercepts
on the graph is summarized in a chart on the next
page.)

Stapel, Elizabeth.
"The Quadratic Formula: Solutions and the Discriminant."
Purplemath. Available from http://www.purplemath.com/modules/quadform2.htm.
Accessed