Quadratic Functions, Quadratic Expressions, Quadratic Equations

Definition: A quadratic function is a function of the form

where a, b, c are real numbers and a 0.

The expression on the right-hand-side

is call a quadratic expression.

cbxaxxf 2)(

cbxax 2

Quadratic Expressions: Factored Form

Examples:

form factored exp. quadratic

)3)(12(352

form factored exp. quadratic

)3)(2(65

2

2

xxxx

xxxx

2.

1.

Factoring quadratic expressions:

Given where a, b, c are integers.

Case 1: a = 1; Since

pqxqpx

pqpxqxx

qxpqxxqxpx

)(

)()())((

2

2

2 cbxax

cbxx 2

we have

pqcqpb

qxpxcbxx

and

ifonly and if

))((2

Examples:

56

127

152

86

2

2

2

2

xx

xx

xx

xx

Case 2: where a, b, c areintegers and a 1.

Since

cbxax 2

pqxpsrqrsx

pqpsxrqxrsxqsxprx

)(

))((2

2

we have

cpqbpsrqars

qsxprxcbxax

,,

ifonly and if

))((2

Examples:

376

5163

352

2

2

2

xx

xx

xx

Quadratic Equations:

A quadratic equation is an equation of the form:

Problem: Find the real numbers x, if any, that satisfy the equation.

The numbers that satisfy the equation are called solutions or roots.

02 cbxax

Methods of Solution:

02 cbxax

Method 1: Factor

then the solutions (roots) of the equationare

cbxax 2

))((If 2 qsxprxcbxax

s

qx

r

px 21 and

Examples:2

2

2 15 0

( 5)( 3) 0

5, 3

2 7 3 0

(2 1)( 3) 0

13,

2

x x

x x

x x

x x

x x

x x

The real number solutions (roots) of the quadratic equation

are:

provided

2 0ax bx c

2 2

1 2

4 4,

2 2

b b ac b b acx x

a a

2 4 0.b ac

Method 2: Use the QUADRATIC FORMULA

The quadratic formula is often written as

2 4

2

b b acx

a

The number is called the discriminant.

2 4b ac

The Discriminant:

Given the quadratic equation

If:

2 0.ax bx c

2

2

2

4 0, the equation has real, unequal roots

4 0, the equation has real, equal roots

4 0, the equation has complex roots

b ac

b ac

b ac

2 4 0b ac (1) ; the roots are:

2 2

1 2

4 4;

2 2

b b ac b b acx x

a a

(2) the roots are: 2 4 0;b ac

1 2 2

bx x

a

(3) no real roots.2 4 0;b ac

Examples:

2

2

1 2

2 7 3 0

( 7) ( 7) 4(2)(3)

2(2)

7 49 24 7 25 7 5

4 4 47 5 7 5 1

3,4 4 2

x x

x

x x

2

2

1 2

3 5 0

3 3 4(1)( 5)

2(1)

3 29

2

3 29 3 29,

2 2

x x

x

x x

2

2 2

3 4 4 0

4 ( 4) 4(3)(4) 16 48 32 0

No real roots.

x x

b ac

Quadratic Functions:

The graph of is a parabola. The graph looks like

if a > 0 if a < 0

2( )f x ax bx c

Key features of the graph:

1. The maximum or minimum point on the graph is called the vertex. The x-coordinate of the vertex is:

2

bx

a

( / 2 ) is the -coordinate of the vertex. f b a y

2. The y-intercept; the y-coordinate of the point where the graph intersects the y-axis.

The y-intercept is:

3. The x-intercepts; the x-coordinates of the points, if any, where the graph intersects the x-axis. To find the x-intercepts, solve the quadratic equation

(0) .f c

2 0.ax bx c

Examples:

Sketch the graph of 2( ) 2 8.f x x x

vertex:

y-intercept:

x-intercepts:

21, (1) 9; vertex (1, 9)

2 2

bx f

a

(0) 8f

2 2 8 ( 4)( 2) 0

4, 2.

x x x x

x x

Sketch the graph of

Vertex:

y-intercept:

x-intercept(s):

2( ) 4 4f x x x

42, (2) 0; (2,0)

2( 1)x f

(0) 4f

2

2

4 4 0

4 4 0; ( 2) 0; 2

x x

x x x x

Sketch the graph of

Vertex:

y-intercept:

x-intercept(s):

2( ) 4 5f x x x

42, (2) 1; (2,1)

2(1)x f

(0) 5f

2 4 5 0 has no real solutions.x x