Quadratic and polynomial forms
Solving quadratic and polynomial equations and inequalities
Contents
![$\displaystyle \begin{pspicture}(-4,0)(4,4)\psset{unit=1cm,arrowsize=8pt}\ps... ...t(2.3,-0.3){\large$x_2$}%\rput(0.35,3.7){\Large$ax^2+bx+c$}\end{pspicture}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img1.png)
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
.
![$ ax^2+bx+c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img2.png)
![$ a$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img3.png)
![$ b$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img4.png)
![$ c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img5.png)
![$ a\not=0$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img6.png)
A number
![$ x$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img105.png)
![$ ax^2+bx+c=0$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img42.png)
![$ ax^2+bx+c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img2.png)
Example: Some quadratic expressions and their coefficients
Quadratic expressions |
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Definition
The number
is called the discriminant
of the quadratic expression
![$\displaystyle \Delta=b^2-4ac .$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img31.png)
![$ ax^2+bx+c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img2.png)
Example: Some quadratic expressions and their discriminant
Quadratic expressions |
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Theorem
If
, the quadratic equation
(where
) has two distinct solutions (also called roots):
et
If
, the quadratic equation
(where
) has a unique solution (or root):
If
, the quadratic equation
has no real solution.
Exercice 1. Solve the following quadratic equations
- a)
- b)
- c)
- d)
Sign of a quadratic expression
Theorem
Let
, (
where
),
then:
![$ f(x)=ax^2+bx+c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img65.png)
![$ a\not=0$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img6.png)
if
, the equation
has two solutions
and
and
-
if
, the equation
has a unique solution
and
-
if
, the quadratic expression
has no root and
Exercice 2. Give the sign of the following quadratic functions:
- a)
- b)
- c)
- d)
Exercices
Exercice 3. Solve the following inequalities:
- a)
- b)
- c)
Exercice 4. Give the sign of the following expression:
- a)
- b)
- c)
Exercice 5. (Examples of equations reducible to quadratic form)
By first defining
![$ X=x^2$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img121.png)
- a)
- b)
Exercice 6. Determine the intersection points (if they exist) between the parabola
![$ \mathcal{P}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img145.png)
![$ \mathcal{D}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img146.png)
![$ \mathcal{P}: y=x^2-3x+1$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img147.png)
![$ \mathcal{D}: y=-2x+1$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img148.png)
Exercice 7. Determine the intersection points (if they exist) between the two parabolas
![$ \mathcal{P}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img145.png)
![$ \mathcal{P}'$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img157.png)
![$ \mathcal{P}: y=x^2-x+2$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img158.png)
![$ \mathcal{P}': y=-x^2+2x-6$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img159.png)
Exercice 8. (A parametric quadratic equation)
Let
![$ m$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img165.png)
![$ 4x^2+(m-1)x+1=0$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img166.png)
Determine the values of
![$ m$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img165.png)
Give then this solution.
Polynomial expressions
Fundamental theorem
Definition
A polynomial is an expression which can be written in the form
where
,
,
,
and
are real numbers, and
is a positive integer.
The integer
is the degree of the polynomial.
A polynomial is an expression which can be written in the form
![$\displaystyle ax^n+bx^{n-1}+cx^{n-2}+\dots+dx+e$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img177.png)
![$ a$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img3.png)
![$ b$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img4.png)
![$ c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img5.png)
![$ d$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img178.png)
![$ e$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img179.png)
![$ n$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img180.png)
The integer
![$ n$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img180.png)
Examples:
![$ \bullet$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img40.png)
![$ P(x)=3x^4-2x^3+\dfrac12 x^2-\sqrt{2}x+3$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img181.png)
![$ \bullet$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img40.png)
![$ Q(x)=5x^7-3x^2+4$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img182.png)
![$ \bullet$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img40.png)
![$ R(x)=x^2+x+1$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img183.png)
Factor Theorem
(Fundamental property for polynomials)
Let
be a polynomial of degree
and
a root of
(that is
).
Then,
can be factored by
:
there exists a polynomial
of degree
such that
Let
![$ P(x)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img77.png)
![$ n$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img180.png)
![$ a$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img3.png)
![$ P$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img184.png)
![$ P(a)=0$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img185.png)
Then,
![$ P(x)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img77.png)
![$ (x-a)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img186.png)
![$ Q(x)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img187.png)
![$ n-1$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img188.png)
![$\displaystyle P(x)=(x-a)Q(x)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img189.png)
Exercice 9. We consider the polynomial function
![$ P(x)=x^3-x^2-x-2$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img190.png)
- Show that
is a root of
, then give a factorization of
.
- Déterminer alors toutes les solutions de l'équation
.
corollary
If the quadratic expression
has two roots
and
,
then it can be factored as
.
If the quadratic expression
![$ ax^2+bx+c$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img2.png)
![$ x_1$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img67.png)
![$ x_2$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img68.png)
![$ ax^2+bx+c=a(x-x_1)(x-x_2)$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img207.png)
Exercice 10. Give a factored expression for the following quadratic expressions.
Exercices
Exercice 11. Let the third degree polynomial
![$ P(x)=2x^3+7x^2+7x+2$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img214.png)
- Show that
is a root of
, then give a factored form of
.
- Solve the equation
, and then give the sign of the expression
.
Exercice 12. Beam deflection
A 2 meter length beam is based on three simple supports
![$ A$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img231.png)
![$ B$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img232.png)
![$ C$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img233.png)
![$ B$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img232.png)
![$ [AC]$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img234.png)
The beams supports a uniformly distributed load of 1000 N.m
![$ ^{-1}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img235.png)
![$\displaystyle %\fbox{\begin{pspicture}(0,0)(11,2.2)\psset{unit=1cm}\pspolyg... ...\rput(4.5,-0.8){$x_m$}\psline[linestyle=dashed](8,-1)(8,-0.2)\end{pspicture}$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img236.png)
One can show that the point located between
![$ B$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img232.png)
![$ C$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img233.png)
![$ x_m$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img237.png)
![$\displaystyle 32x^3-156x^2+240x-116=0 .$](/Lycee/Common/Cours-2nd-degre/Cours-2nd-degre/img238.png)
- 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 ?
![Lien vers les devoirs corrigés](/include/fleche_Droite.png)