| |
|
|
|
|
|
|
|
|
|
|
| Algebraic Equations and Inequalities |
 Properties of Equations and Inequalities Graphical Interpretation of Solutions
Linear Equations Linear Inequalities
Linear Equations Involving Absolute Value Linear Equations Involving a Few Absolute Values Linear Inequalities Involving Absolute Value
 Quadratic
EquationsQuadratic Equations and Quadratic Functions Extreme Value of Quadratic Function Quadratic Formula Solving Quadratic Equations by Factoring Quadratic Inequalities |
|
|
A quadratic inequality can be put into one of the following forms, a x2
+ b x + c > 0
or
a x2
+ b x + c ≥ 0,
where x
is the variable; a , b
and c are constants (a
≠ 0).
In order to solve a quadratic inequality, it is necessary to solve the corresponding quadratic equation, x2
+ b x + c = 0.
The solution set for the inequality depends on the sign of the determinant D. If D < 0, then the above equation has no real roots, and hence, the expression x2
+ b x + c
holds its sign as is shown in the figure below.
If a > 0 then x2
+ b x + c > 0
for any x, and so the solution set consists of all real numbers. If a < 0 then x2
+ b x + c < 0
for any x, and so the solution set is an empty set.
If D < 0, then the quadratic equation has no real roots If x1 and x2 are the roots of the equation, then the quadratic polynomial can be factored into two linear factors, x2
+ b x + c = (x
– x1)(x
– x2).
Expand the expression on the right side: x2
+ b x + c = x2
– x (x1
+ x2)
+ x1 x2.
Regroup the terms: (b
+ x1 + x2)
x + c = x1
x2.
This identity has to be true for any values of x.
c
= x1 x2. (*)
Therefore, (b
+ x1 + x2)
x = 0
for any values of x.
Hence,
x1 + x2
= – b. (**)
Formulas (*) and (**) can be used to find the roots of a quadratic equation as well as to check up whether some supposed roots are correct.
|
