# Quadratic and polynomial forms

## Solving quadratic and polynomial equations and inequalities

Contents

### 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 .

Example: Some quadratic expressions and their coefficients

Definition The number is called the discriminant of the quadratic expression

Example: Some quadratic expressions and their discriminant

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.

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

Exercice 9. We consider the polynomial function .
1. Show that is a root of , then give a factorization of .

2. Déterminer alors toutes les solutions de l'équation .

corollary
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 .
1. Show that is a root of , then give a factored form of .

2. 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:

1. Verify that is a solution of this equation.

2. Give a factored form of the polynomial expression of this equation and then solve it.

3. Find , the location between points and , where the beam is at most deformed.

Some more exercices on quadratic forms ?