Transcript Section 3 - Juan Diego Academy

CHAPTER 2
2-3 solving quadratic equations by graphing and
factoring
Objectives
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Students will be able to:
Solve quadratic equations by graphing or
factoring.
Determine the quadratic function from its roots.
Zero of a function
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What is the zero of a function?
A zero of a function is a value of
the input x that makes the output
f(x) equal zero. The zeros of a
function are the x-intercepts.
Solutions of a quadratic function
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How many solutions or zeros does a quadratic
function has?
Unlike linear functions, which have
no more than one zero, quadratic
functions can have two zeros, as
shown at next page. These zeros are
always symmetric about the axis of
symmetry
Solutions of a quadratic function
Example 1
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Find the zeros of f(x) = x2 – 6x +
8 by using a graph and table.
Example 1 solution
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The table and the graph indicate that
the zeros are 2 and 4.
Example 2
Find the zeros of g(x) = –x2 – 2x
+ 3 by using a graph and a table.
2
 Enter y = –x – 2x + 3 into a
graphing calculator.
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Example 2 solution
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Both the table and the graph show
that y = 0 at x = –3 and x = 1.
These are the zeros of the function.
Student Guided practice
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Lets do the quadratic equations worksheet
The roots of an equation
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Besides zeros what is another name for the solutions
of a quadratic equation?
The solution to a quadratic equation
of the form ax2 + bx + c = 0 are
roots. The roots of an equation
are the values of the variable that
make the equation true.
Factoring quadratic equations
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Another way we can find the solution of a quadratic
equation is called factoring.
You can find the roots of some
quadratic equations by factoring and
applying the Zero Product Property.
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Remember:
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Functions have zeros or x-intercepts.
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Equations have solutions or roots.
Zero product property
Example 3
Find the zeros of the function by
factoring.
 f(x) = x2 – 4x – 12
 Solution:
 first we set up the equation =0.
 𝑥2 – 4𝑥 – 12 = 0.
 Second we distribute the x’s
 (𝑥
) (𝑥 )
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Example 3 continue
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Third we factor the coefficient and we think of two
numbers that multiply we get the last number and
add those two numbers we get the middle number.
(𝑥 + 2)(𝑥 – 6) = 0
 then we solve for x both factors
 𝑥 + 2 = 0 𝑜𝑟 𝑥 – 6 = 0
 x= –2 or x = 6
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Example 4
find the solution to the equation
𝑦 = 3𝑥 2 + 5𝑥 − 12
Binomials and trinomials
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Quadratic expressions can have one,
two or three terms, such as –16t2, –
16t2 + 25t, or –16t2 + 25t + 2.
Quadratic expressions with two
terms are binomials. Quadratic
expressions with three terms are
trinomials. Some quadratic
expressions with perfect squares
have special factoring rules.
Special rule
Example 5
Find the roots of the equation by
factoring.
2
 18x = 48x – 32
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Example 6
Find the roots of the equation by
factoring.
2
 x – 4x = –4
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Example 6
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Solve the equation by factoring.
(k + 1)(k − 5) = 0
Student guided practice
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Lets do problems from worksheet 2-6
Example 7
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Write a quadratic function in
standard form with zeros 4 and –
7.
Example 8
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Write a quadratic function in
standard form with zeros 5 and –
5.
Homework
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Do problems 2-10, 12,15 from page 82 in the book
closure
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Today we saw how we can solve quadratic
equations by graphing and factoring.