LESSON 3 - 6 : The Zeros of a Quadratic Function
MCR3U1
(Nature of the Roots)
MINDS ON...
The demand to create automotive parts is increasing.
BMW developed three different methods to develop these parts.
The profit function for each method is given below, where
y is the profit and x is the quantity of parts sold in thousands:
PROCESS A:
PROCESS B:
PROCESS C:
P(x) = -0.5x2 + 3.2x –5.12
P(x) = -0.5x2 + 4x – 5.12
P(x) = -0.5x2 + 2.5x – 3.8
The graphs of the corresponding profit functions are shown below.
PROCESS A
PROCESS B
PROCESS C
Which process would you recommend? Explain.
LESSON 3 - 6 : The Zeros of a Quadratic Function
MCR3U1
(Nature of the Roots)
Recall: y = ax2 + bx + c
0 = ax2 + bx + c

is a Quadratic Function (a relation between x and y)
is a Quadratic Equation (let y = 0 to find the roots and ZEROS)
−𝒃±√𝒃𝟐 −𝟒𝒂𝒄
2
The roots of the quadratic equation ax + bx + c = 0 are
where the
=
radicand,

𝟐
𝒃 − 𝟒𝒂𝒄 is called the DISCRIMINANT, D.
𝟐𝒂
,
The value of the discriminant determines the number and nature the roots of a quadratic
equation (that is, real/not, equal/distinct) and the number of x-intercepts.
Investigation
Complete the table below and observe the value of the discriminant, D, in each case.
Quadratic
Equation
Function in
Factored Form
Function in
Vertex Form
x 2  4 x  12  0
x 2  6x  9  0
x 2  2x  3  0
Graph of the
Quadratic
Function
Root(s)
(Use the
Quadratic
Formula)
Nature of Roots
(real/not real,
equal/distinct)
Sign of
Discriminant
Summary:
D = b2 – 4ac
D=
If D= b2 – 4ac > 0
D = b2 – 4ac
D=
If D = b2 – 4ac = 0
D = b2 – 4ac
D=
If D = b2 – 4ac < 0
Note: A quadratic equation may have 0, 1 or 2 solutions, but only a maximum of 2.
Three Possibilities exist:
Ex. 1: Determine the number and nature of the zeros for the quadratic equations without solving.
(i.e., Does the parabola intersect the x-axis at one point, two points or not at all?)
a) x2 + 5x – 8 = 0
b) 3x2 + 2x + 7 = 0
Ex. 2: Use the vertex and the direction of opening to determine the number of zeros of the
function.
a) f(x) = -2(x + 3)2 – 4
b) f(x) = -3(x + 2)2 + 4
Ex. 3: For what value(s) of k will the quadratic equation x2 + kx + 25 = 0 have:
a) two distinct solutions
b) one real solution
c) no real solution
Homefun: p. 185 #1-3 odds, 5ac, 6, 8, 9, 10, 14 (Note: to break-even, Profit = 0)
LESSON 3 - 6 : The Zeros of a Quadratic Function
MCR3U1
(Nature of the Roots)
Recall: y = ax2 + bx + c
0 = ax2 + bx + c

is a Quadratic Function (a relation between x and y)
is a Quadratic Equation (let y = 0 to find the roots and ZEROS)
−𝒃±√𝒃𝟐 −𝟒𝒂𝒄
2
The roots of the quadratic equation ax + bx + c = 0 are
where the
=
radicand,

,
𝟐𝒂
𝟐
𝒃 − 𝟒𝒂𝒄 is called the DISCRIMINANT, D.
The value of the discriminant determines the number and nature the roots of a quadratic
equation (that is, real/not, equal/distinct) and the number of x-intercepts.
Investigation
Complete the table below and observe the value of the discriminant, D, in each case.
Quadratic
Equation
Function in
Factored Form
Function in
Vertex Form
Graph of the
Quadratic
Function
Root(s)
Use the
Quadratic
Formula
y  x 2  4 x  12
y  x 2  6x  9
y  x2  2x  3
y = (x+6)(x-2)
y = (x – 3)(x – 3)
Cannot Be Factored
y  x  2  16
y   x  3
y  ( x  1) 2  2
2
𝒙=
𝒙=
−𝒃±√𝒃𝟐 −𝟒𝒂𝒄
𝟐𝒂
𝒙=
−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
𝒙=
−𝟒 ± √𝟒𝟐 − 𝟒(𝟏)(−𝟏𝟐)
𝟐(𝟏)
𝒙=
−(−𝟔) ± √(−𝟔)𝟐 − 𝟒(𝟏)(𝟗)
𝟐(𝟏)
𝒙=
−(−𝟐) ± √(−𝟐)𝟐 − 𝟒(𝟏)(𝟑)
𝟐(𝟏)
𝒙=
𝟐 ± √−𝟖
𝟐
𝒙=
−𝟒 ± √𝟔𝟒
𝟐
x = 2, x = -6
Nature of Roots
(equal/distinct,
real/not real)
Sign of
Discriminant
Summary
2
Two Distinct Real
Solutions
D = b2 – 4ac
D = (4)2 – 4(1)(-12)
D = +ve
If D > 0
TWO DISTINCT REAL
ROOTS and
X-INTERCEPTS
𝟔 ± √𝟎
𝒙=
𝟐
x = 3, x = 3
Two Equal Real Solutions
D = b2 – 4ac
D = (-6)2 – 4(1)(9)
D=0
If D = 0
TWO EQUAL REAL ROOTS
and
1 X-INTERCEPT
−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
x  1  i 2, x  1  i 2
Two Distinct Imaginary
Solutions
(NO REAL ROOTS)
D = b2 – 4ac
D = (-2)2 – 4(1)(3)
D = -ve
If D < 0
NO REAL ROOTS
and
NO X-INTERCEPTS