Transcript Quadratic Equations and Problem Solving Lesson 3.2 Finding Zeros  Often with quadratic functions f(x) = a*x2 + bx + c we speak of “finding the.

Quadratic Equations
and Problem Solving
Lesson 3.2
Finding Zeros

Often with quadratic functions
f(x) = a*x2 + bx + c
we speak of “finding the zeros”

This means we wish to find all possible values
of x for which
a*x2 + bx + c = 0
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Finding Zeros



Another way to say this is that we are seeking the xaxis intercepts
This is shown on the graph below
Here we see two zeros – what other possibilities
exist?
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Zeros of the Quadratic

Zeros are where the function crosses the x-axis


Where y = 0
Consider possible numbers of zeros

None (or two complex)
One
Two
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Factoring



Given the function x2 - 2x - 8 = 0
Factor the left side of the equation
(x - 4)(x + 2) = 0
We know that if the product of two numbers
a * b = 0 then either ...



a=0
b=0
or
Thus either
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
x - 4 = 0 ==> x = 4 or
x + 2 = 0 ==> x = -2
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Warning!!


Problem ... many (most) quadratic functions
are NOT easily factored!!
Example:
f ( x)  3x  7 x  7
2
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Completing the Square



We work with a quadratic
equation to make one side a
perfect square
Then we take the square root of
both sides
Not forgetting to use both the +
and - values of the right side of
the equation
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The Quadratic Formula

We can use
completing the
square with the
general equation
Once this is done, we can
use
2 the formula for any
quadratic function.
ax + bx + c = 0.
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The Quadratic Formula



It is possible to create two functions on your
calculator to use the quadratic formula.
quad1 (a,b,c)
which uses the -b + ...
quad2 (a,b,c)
which uses the -b - ...
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The Quadratic Formula

Try it for the quadratic functions


4x2 - 7x + 3 = 0
6x2 - 2x + 5 = 0
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The Quadratic Formula
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4x2 - 7x + 3 = 0
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The Quadratic Formula

Why does the second function give "non-real
result?“

6x2 - 2x + 5 = 0
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The Discriminant

Consider the expression under the radical in the
quadratic formula
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b  b  4ac
2a


This is known as the discriminant
What happens when it is




Positive and a perfect square?
Positive and not a perfect square?
Zero
Negative?
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Graphical Solution

Given




x 2  3.1x  0.32
Manipulate the equation to be equal to zero
Specify this as a function of x on Y= screen
Graph and note zeros
Use F5 menu
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Numeric Solution

Given



x 2  3.1x  0.32
As before …
Manipulate the equation to be equal to zero
Specify this as a function of x on Y= screen
Now go to the Table, use ♦Y


Look for x-value where y-values go from negative to
positive
Use setup, F2 to change
start and increment
to "zoom in" on the numeric
answer
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Assignment



Lesson 3.2
Page 200
Exercises 1 – 77 EOO
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