Transcript Document

OCF.01.5 - Finding Zeroes of Quadratic Equations
MCR3U - Santowski
1
(A) Review

Zeroes is another term for roots or x-intercepts - basically, the point
where the function crosses the x-axis. At this point, the y value of the
function is 0.

A quadratic may have one of the following three possibilities : 2
distinct zeroes, one zero (the x-intercept and the vertex are one and
the same), or no zeroes (the graph does not cross the x-axis)

See diagrams on the next slide
2
(A) Review – Zeroes of Quadratics

Diagrams of each scenario (0,1,2 zeroes)
3
(B) Finding the Zeroes

The zeroes of a quadratic function can be found in a variety of ways:

(i) Factoring: If a quadratic equation can be factored to the form of y
= a(x - s)(x - t), then the zeroes are at (s,0) and (t,0)

ex: Find the roots of y = 4x2 - 12x + 9

(ii) Completing the Square technique: If an equation can be written
in the y = a(x – h)2 + k form, then the (x – h)2 term can be isolated in
order to solve for x

ex 1. Find the roots of y = x2 - 6x - 27 by using the method of
completing the square
ex 2. Solve y > 2x2 - 5x - 1 using the completing the square
technique

4
(B) Graph of y > 2x2 – 5x – 1
5
(B) Finding the Zeroes

(iii) The Quadratic Formula:

For an equation in the form of y = ax2+ bx+c, then the quadratic
formula may be used:
x = [- b + (b2-4ac)] / 2a


ex 1. Find the roots of the y = x2 - 2x - 3 using the quadratic formula
ex 2. Solve y < -2x2 + 5x + 8 using the quadratic formula

(iv) Using a Graphing Calculator/Technology

ex 1. Graph y = 4x2 + 8x - 24 and find the intercepts.
ex 2. Solve y < 1/4x2 + 5x – 9 using the GDC


6
(C) The Discriminant

You can use part of the quadratic formula, the discriminant (b2 - 4ac)
to predict the number of roots a quadratic equation has.

If b2 - 4ac > 0, then the quadratic equation has two zeroes

ex: y = 2x2 + 3x – 6

If b2 - 4ac = 0, then the quadratic equation has one zero
ex: y = 4x2 + 16x + 16



If b2 - 4ac < 0, then the quadratic equation has no zeroes
ex: y = -3x2 + 5x - 3
7
(D) Visualizing and Interpreting The Discriminant
8
(E) Interpretation of Zeroes

ex 1. The function h = -5t2 + 20t + 2 gives the approximate height, in
meters, of a thrown football as a function of time in seconds. The
ball hit the ground before the receiver could get there.

(a) For how long was the ball in the air?
(b) For how many seconds was the height of the ball at least 10
meters?

9
(F) Internet Links
 College Algebra
Tutorial on Quadratic
Equations
 Solving Quadratic Equations Lesson - I
from Purple Math
10
(D) Homework


Handout from MHR, p129, Q5bce, 6g, 7, 8ab, 9acegk, 11ace,
12egmn
Nelson Textbook, p325, Q1-5eol, 7-11,6
11