Quadratic and polynomial forms
Solving quadratic and polynomial equations and inequalities
Contents
Quadratic form
Solving quadratic equations
Definition
A quadratic expression is an expression which can be written as
, where
,
et
represent numbers such that
.
A number solution of the quadratic equation is also called a root of the quadratic expression .
A number solution of the quadratic equation is also called a root of the quadratic expression .
Example: Some quadratic expressions and their coefficients
Quadratic expressions  
Definition
The number
is called the discriminant
of the quadratic expression
Example: Some quadratic expressions and their discriminant
Quadratic expressions  
Theorem

If
, the quadratic equation
(where ) has two distinct solutions
(also called roots):
et  If , the quadratic equation (where ) has a unique solution (or root):
 If , the quadratic equation has no real solution.
Exercice 1. Solve the following quadratic equations
 a)
 b)
 c)
 d)
Sign of a quadratic expression
Theorem
Let
, (
where
),
then:

if
, the equation
has two solutions
and
and

if
, the equation
has a unique solution
and

if , the quadratic expression
has no root and
Exercice 2. Give the sign of the following quadratic functions:
 a)
 b)
 c)
 d)
Exercices
Exercice 3. Solve the following inequalities:
 a)
 b)
 c)
Exercice 4. Give the sign of the following expression:
 a)
 b)
 c)
Exercice 5. (Examples of equations reducible to quadratic form)
By first defining , solve the following equations:
 a)
 b)
Exercice 6. Determine the intersection points (if they exist) between the parabola and the line which equations are: et
Exercice 7. Determine the intersection points (if they exist) between the two parabolas and where: et
Exercice 8. (A parametric quadratic equation)
Let be a real number. We consider the quadratic equation .
Determine the values of for which this equation has a unique solution.
Give then this solution.
Polynomial expressions
Fundamental theorem
Definition
A polynomial is an expression which can be written in the form
where ,
,
,
and
are real numbers, and
is a positive integer.
The integer is the degree of the polynomial.
A polynomial is an expression which can be written in the form
The integer is the degree of the polynomial.
Examples:
is a polynomial of degree 4.
is a polynomial of degree 7.
is a polynomial of degree 2. This is also a quadratic expression: actually all quadratic expression are second degree polynomial.
Factor Theorem
(Fundamental property for polynomials)
Let be a polynomial of degree and a root of (that is ).
Then, can be factored by : there exists a polynomial of degree such that
Let be a polynomial of degree and a root of (that is ).
Then, can be factored by : there exists a polynomial of degree such that
Exercice 9. We consider the polynomial function .
 Show that
is a root of
,
then give a factorization of
.
 Déterminer alors toutes les solutions de l'équation
.
corollary
If the quadratic expression has two roots and , then it can be factored as .
If the quadratic expression has two roots and , then it can be factored as .
Exercice 10. Give a factored expression for the following quadratic expressions.
Exercices
Exercice 11. Let the third degree polynomial .
 Show that
is a root of
, then give a factored form of
.
 Solve the equation
,
and then give the sign of the expression
.
Exercice 12. Beam deflection
A 2 meter length beam is based on three simple supports , and , supporting point being located in the middle of the segment .
The beams supports a uniformly distributed load of 1000 N.m (newtons metre). Under the action of this charge, the beam deforms.
One can show that the point located between and where the deformation is maximum, is such that is a solution of the equation:
 Verify that
is a solution of this equation.
 Give a factored form of the polynomial expression of this equation and then solve it.
 Find
,
the location between points
and
,
where the beam is at most deformed.
Some more exercices on quadratic forms ?