GMAT Quantitative Review
3.0 Math Review
3.2 Algebra
6. Solving Quadratic Equations
The standard form for a quadratic equation is
,
where a, b, and c are real numbers and ; for example:
Some quadratic equations can easily be solved by factoring. For example:
(1) |
|
(2) |
|
A quadratic equation has at most two real roots and may have just one or even no real root. For example, the equation can be expressed as
, or
; thus the only root is 3. The equation
has no real root; since the square of any real number is greater than or equal to zero,
must be greater than zero.
An expression of the form can be factored as
.
For example, the quadratic equation can be solved as follows.
If a quadratic expression is not easily factored, then its roots can always be found using the quadratic formula: If
, then the roots are
These are two distinct real numbers unless . If
, then these two expressions for x are equal to
, and the equation has only one root. If
, then
is not a real number and the equation has no real roots.