Zeros of Quadratic Function

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: P(x) = -0.5x2 + 3.2x –5.12
PROCESS B: P(x) = -0.5x2 + 4x – 5.12
PROCESS C: 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 is a Quadratic Function (a relation between x and y) 0 = ax2 + bx + c is a Quadratic Equation (let y = 0 to find the roots and ZEROS)
The roots of the quadratic equation ax2 + bx + c = 0 are , where the radicand, 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)

Nature of Roots
(real/not real, equal/distinct)

Sign of Discriminant
D = b2 – 4ac
D =

D = b2 – 4ac
D =
D = b2 – 4ac
D =
Summary:

If D= b2 – 4ac > 0

If D = b2 – 4ac = 0

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 + 7 = -2x

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) one real solution b)two distinct solutions 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 is a Quadratic Function (a relation between x and y) 0 = ax2 + bx + c is a Quadratic Equation (let y = 0 to find the roots and ZEROS)
The roots of the quadratic equation ax2 + bx + c = 0 are , where the radicand, 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 y = (x+6)(x-2) y = (x – 3)(x – 3)
Cannot Be Factored
Function in Vertex Form

Graph of the
Quadratic
Function

Root(s)

Use the
Quadratic Formula

x = 2, x = -6

x = 3, x = 3

Nature of Roots (equal/distinct, real/not real)
Two Distinct Real Solutions
Two Equal Real Solutions
Two Distinct Imaginary Solutions
(NO REAL ROOTS)
Sign of Discriminant
D = b2 – 4ac
D = (4)2 – 4(1)(-12)
D = +ve
D = b2 – 4ac
D = (-6)2 – 4(1)(9)
D = 0
D = b2 – 4ac
D = (-2)2 – 4(1)(3)
D = -ve
Summary

If D > 0
TWO DISTINCT REAL ROOTS and X-INTERCEPTS
If D = 0
TWO EQUAL REAL ROOTS and
1 X-INTERCEPT
If D < 0
NO REAL ROOTS and NO X-INTERCEPTS