This is a free lesson from our course in Algebra II
In this section you'll explore with the help of several examples how to classify
the roots using the discriminant from the quadratic formula. You may find it in the beginning to be
elusive but walking through some examples with solution; and explanation by the
instructor with the help of audio, video presentation in own handwriting; it may
be simpler to understand. In the quadratic equation ax2
+ bx + c, the expression underneath the square root; sign i.e. b2 - 4ac
is called the discriminant of the quadratic equation. It can be used to determine the condition for real, equal, or rational roots.A quadratic equation may have two
(real) roots, one root or no roots. It does not have more than two roots, except
in the situation if a = b = c = 0, where the expression
exists for all values of x.
(More text below video...)
(Continued from above)
The roots are given by
x = {-b
(b2 - 4ac)}/2a.
The conditions for the roots to be real, imaginary, rational, irrational, equal or unequal are as follows:
•
if b2 - 4ac < 0, gives 2 conjugate
imaginary roots .
•
if b2 - 4ac = 0, gives 1 real, rational
and equal root.
•
if b2 - 4ac > 0 and a perfect square, gives 2 real, rational and unequal
root. If
b2 - 4ac > 0 and not a perfect square, then it gives 1 real, irrational and unequal root. For example, the quadratic equation y
= x2 - 10x + 26 have two imaginary roots, since b2
- 4ac = -4 i.e. < 0.1 Note: When using the Quadratic Formula, make sure not to omit the '±'sign, and be careful with the fraction line (it's under the initial b part, too).
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(b2 - 4ac)}/2a.