# Equations and inequations of the second degree: 11th grade math lesson

The equations and inequations of the second degree in a math lesson for 11th grade where we will approach the resolution with the delta discriminant and the factorization of a polynomial of the second degree as well as the study of its sign. In this first lesson, we will study graphical interpretation.

In all this chapter, we will consider a a non-zero real.

## I. Solving the second degree equation :

### 1. Definition and vocabulary :

Definition:
1. A second degree equation, with one unknown x, is an equation that can be written as , where are three real numbers given with .
2. Solving the equation , is to find all the numbers such that .
3. Such a number p is called the solution or root of the equation.

### 2. Solving the second degree equation :

Let f(x) = ax²+bx+c with .

#### 2.1. Writing f(x) in canonical form:

Definition:

Since , or

so .

The latter is called the canonical form of f.

#### 2.2. Solving the equation ax²+bx+c=0 :

Ownership:

We pose as follows

Case 1:

If then .

The number in square brackets is strictly positive so the equation f(x)=0 has no solution.

Second case:

If then .

Since , the equation f(x)=0 has one and only one solution:

.

Third case:

If then and :

If we pose :

and then .

So since , the equation f(x)=0 has two distinct solutions and .

Definition:

The number is called the discriminant of the second degree equation or the trinomial .

Theorem:

a. When , the equation has no solution in .

b. When , the equation has a double root: .

c. When , the equation has two solutions:

and .

## II. Factorisation and sign of the trinomial :

### 1. Factorisation of the trinomial :

We have seen in the proof of Theorem 1 that if

then .

Theorem 2 : factorization of the trinomial.

When the equation f(x)=ax²+bx+c=0 has two solutions and ( in the case ) then,

### 2. Sign of the trinomial :

Theorem
1. When , is always of the sign of a.
2. When , has the sign of a
3. When , has the sign of a, except when x is between the roots, in which case f(x) and a have opposite signs.

Application:

To solve a second degree inequation, we determine the sign of the associated trinomial.

## III. Graphical representations of trinomial functions :

Definition:

The curve of the function is a parabola. This parabola faces upwards when and faces downwards when .

Synopsis:

Examples:

Solve

Solution:

since , the trinomial has no root in .

Moreover a=1 so a>0 so for any real and .

Solve the second degree inequation .

We have .

The equation has two roots which are t:

= 1 and .

We have so the solution set of the second degree inequation is the interval .

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