Transcript (a = 1). - Cloudfront.net

8-3A Factoring Trinomials and
Solving Quadratic Equations
There are numerous methods to factor trinomials.
The method used in this presentation is NOT in your
textbook. Please pay attention as this method is
easier to use than the method presented in the book!
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
In the previous lesson, you solved a quadratic equation
by factoring.
The problem.
Set each factor equal
to zero and solve!
x  3x  5  0
x 3  0
x  3
or
x5  0
or
x  5
The factors were given information. Today you will
need to find the factors of a quadratic trinomial and
then use the factors to solve a quadratic equation.
Quadratic expressions are written in the following way:
ax2  bx  c
quadratic
trinomial
leading
coefficient
Today we will factor trinomials when the leading
coefficient is 1.
x 2  bx  c
The coefficient of x 2 is 1.
When the coefficient of ax 2 is 1.
ax2  bx  c becomes
x 2  bx  c
ac
Product ac
b
Sum b
(a = 1).
To factor the trinomial we will.
Multiply a times c and place this on top.
And place b in the bottom.
To fill in the sides of the x you must find two
numbers that have a product of ac and a sum of b.
Factor.
Multiply a times c to find the product.
ax2  bx  c
15
x2  8x
8  15  5  3
Draw an X
on your
x
x

b in the bottom represents
the sum
paper.
To fill in the sides of the x you must find two
This
quadratic
numbers that have
a product
of trinomial
15 and a sum of 8.
is an
expression.
Place the values from
the
sides of theHow
X figure into
do you know it is NOT
your factors.
and equation?
Check by doing
You know
FOIL in your
there will be
head!
an x in each
factor!
All of today’s problems involving quadratic trinomials will
have a leading coefficient of 1.
Multiply a times c to find the product.
–3
ax2  bx  c
x2 
–22x  3 –33  1
x x 
b in the bottom represents the sum
To fill in the sides of the x you must find two
numbers that have a product of –3 and a sum of –2.
Place the values from the sides of the X figure into
your factors.
Check by doing
You know
FOIL in your
there will
head!
be an x in
each factor!
Example 1 Factor.
1. Write the problem.
2. Draw an X next to
the problem.
ax2  bx  c
x2  7x  12
x x 
12
+4
4 +3
3
7
3. Multiply a times c to find the product.
4. Write b in the bottom to represent the sum.
5. Fill in the sides of the x by finding two
numbers that have a product of the top
Check by
number and a sum of the bottom number.
doing FOIL
6. Using the values from the sides of
in your
the X figure write the factors. Your
head!
factors must be in parentheses!
Solve.
1) x 2  4 x  3  0
x 1x  3  0
x + 1 = 0 or x + 3 = 0
- 1 -1 or - 3 -3
x = -1 or
x=-3
-1 and -3 are the x-intercepts (roots, zeros or solutions of
the quadratic function:
y  x2  4x  3
Recall that to find the x-intercepts, you let y = 0.
You do the same for a quadratic function. This will graph
a PARABOLA!
Axis of symmetry
x=
= -4 = -2
2(1)
Quadratic Function
y = ax2 + bx + c
y  x2  4x  3
Let x = 0
y-intercept
( 0, c )
(0,3)
Quadratic Equation
ax2 + bx + c = 0
x2  4x  3  0
Vertex
(
x-intercept’s
-1, -3
, y)
Let y = 0
Factor and solve
x2  4x  3  0
(x +1) (x + 3) = 0
(-2, -1)
a= 1 b= 4
c= 3
a>0 so parabola opens up
Vertex : at the vertex, x=-2
Substitute x=-2 into the quadratic function
y  x2  4x  3
Vertex = (-2, -1)
y = (-2)2 + 4(-2) + 3 = -1
Vertex = (-2, -1)
8-A6 Page 438 # 12–31.