2-3
Solving Quadratic Equations by
Graphing and Factoring
Warm Up
Find the x-intercept of each function.
1. f(x) = –3x + 9
2. f(x) = 6x + 4
Factor each expression.
3. 3x2 – 12x
5. x2 – 49
Holt McDougal Algebra 2
4. x2 – 9x + 18
2-3
Solving Quadratic Equations by
Graphing and Factoring
Objectives
Solve quadratic equations by graphing or
factoring.
Determine a quadratic function from its
roots.
Vocabulary
zero of a function
root of an equation
binomial
trinomial
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
x2  bx  c
2
x  bx  c
 x  3 x  5 
8x is the sum of the O and I part of FOIL.
x  bx  c
x2  bx  c
2
 x  r  x  s 
For “c” to be negative, “r” and “s” must have opposite signs
 x  5 x  3 
Holt McDougal Algebra 2
Solving Quadratic Equations by
Graphing and Factoring
2-3
Factor the trinomial
1. r  14r  45
2. a 2  9a  20
3. k 2  23k  24
4. x2  2 x  63
5.  24  5x  x2
6. r 2  4r  21
2
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Same general idea for factoring
2
but we need to look for the factors
ax  bx  c
of “ac” that combine to make “b”.
9. 3x2  5 x  2
10. 2 x 2  21x  11
11. 8x2  14 x  15
Holt McDougal Algebra 2
12. 6 x  5 x  21
2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Special Cases
Difference of Squares – Product of a sum and
2
2
a

b
  a  b  a  b 
a difference.
Perfect Square Trinomials – The product of the
square of a sum or the square of a difference.
a  2ab  b   a  b 
2
2
a 2  2ab  b2   a  b 
Holt McDougal Algebra 2
2
2
Solving Quadratic Equations by
Graphing and Factoring
2-3
1. x 2  25
2. a 2  121
2
5.
49
v
 56v  16
4. 4 s  36 s  81
2
Holt McDougal Algebra 2
3. 16 x 2  1
6. 9  16 g 2  24 g
2-3
Solving Quadratic Equations by
Graphing and Factoring
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.
Unlike linear functions,
which have no more
than one zero,
quadratic functions can
have two zeros, as
shown at right. These
zeros are always
symmetric about the
axis of symmetry.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
You can find the roots of some quadratic equations by
factoring and applying the Zero Product Property.
Reading Math
• Functions have zeros or x-intercepts.
• Equations have solutions or roots.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Example 1: Finding Zeros by Using a Graph or Table
Find the zeros of f(x) = x2 – 6x + 8 by using a
graph and table.
Method 1 Graph the function f(x) = x2 – 6x + 8.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Example 1 Continued
Find the zeros of f(x) = x2 – 6x + 8 by using a
graph and table.
Method 2
Use a calculator.
Enter y = x2 – 6x + 8 into a graphing calculator.
Both the table and the graph show that y = 0 at
x = 2 and x = 4. These are the zeros of the function.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Any object that is thrown or launched into the air,
such as a baseball, basketball, or soccer ball, is a
projectile. The general function that approximates
the height h in feet of a projectile on Earth after
t seconds is given.
Note that this model has limitations because it
does not account for air resistance, wind, and
other real-world factors.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Example 3: Sports Application
A golf ball is hit from ground level with an
initial vertical velocity of 80 ft/s. After how
many seconds will the ball hit the ground?
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
Example 3 Continued
Check The graph of the function h(t) = –16t2 + 80t
shows its zeros at 0 and 5.
105
–3
7
–15
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
If you know the zeros of a function, you can
work backward to write a rule for the function
Example 5: Using Zeros to Write Function Rules
Write a quadratic function in standard form
with zeros 4 and –7.
Holt McDougal Algebra 2
2-3
Solving Quadratic Equations by
Graphing and Factoring
HW Day 1 wkst 1-30
Day 2 wkst 31-52
Day 3 18 – 58 even, 27,47,67 - 69
Holt McDougal Algebra 2