5.2.2 Solve by Factoring - Social Circle City Schools

Introduction
Recall that a factor is one of two or more numbers or
expressions that when multiplied produce a given
product. We can factor certain expressions by writing
them as the product of factors.
The Zero Product Property states that if the product of
two factors is 0, then at least one of the factors is 0. After
setting a quadratic equation equal to 0, we can
sometimes factor the quadratic expression and solve the
equation by setting each factor equal to 0.
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5.2.2: Factoring
Key Concepts
• The greatest common factor, or GCF, is the largest
factor that two or more terms share.
• You should always check to see if the terms of an
expression have a greatest common factor before
attempting to factor further.
• The value of a for a quadratic expression in the form
ax2 + bx + c is called the leading coefficient, or lead
coefficient, because it is the coefficient of the term
with the highest power.
2
5.2.2: Factoring
Key Concepts, continued
• To factor a trinomial with a leading coefficient of 1 in
the form x2 + bx + c, find two numbers d and e that
have a product of c and a sum of b.
• The factored form of the expression will be
(x + d)(x + e).
• When finding d and e, be careful with the signs.
• The table that follows shows what the signs of d and e
will be based on the signs of b and c.
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5.2.2: Factoring
Key Concepts, continued
Signs of b, c, d, and e
b
c
d
e
+
+
+
+
–
+
–
–
+
–
–
Opposite signs; the number
with the larger absolute
value is positive.
–
Opposite signs; the number
with the larger absolute
value is negative.
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5.2.2: Factoring
Key Concepts, continued
• You may be able to factor expressions with lead
guess-and-check.
• Expressions with lead coefficients other than 1 in the
form ax2 + bx + c can sometimes be factored by
grouping.
• If you struggle to factor expressions in your head or by
using guess-and-check, factoring by grouping is a
more structured alternative.
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5.2.2: Factoring
Key Concepts, continued
Factoring by Grouping
1. Begin by finding two numbers d and e whose
product is ac and whose sum is b.
2. Rewrite the expression by replacing bx with dx + ex:
ax2 + dx + ex + c.
3. Factor the greatest common factor from ax2 + dx.
4. Factor the greatest common factor from ex + c.
5. Factor the greatest common factor from the
resulting expression.
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5.2.2: Factoring
Key Concepts, continued
• A quadratic expression in the form (ax)2 – b2 is called
a difference of squares.
• The difference of squares (ax)2 – b2 can be written in
factored form as (ax + b)(ax – b).
• Some expressions cannot be factored. These
expressions are said to be prime.
• Although the difference of squares is factorable, the
sum of squares is prime.
• For example, (3x)2 – 52 = 9x2 – 25 = (3x + 5)(3x – 5),
but (3x)2 + 52 = 9x2 + 30x + 25 and is not factorable.
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5.2.2: Factoring
Common Errors/Misconceptions
• forgetting to consider the signs of a, b, and c
• treating a lead coefficient other than 1 as if it were a 1
• multiplying the terms of an expression given in factored
form to solve an equation
• solving an expression that is not part of an equation
• confusing the difference of squares with the sum of
squares
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5.2.2: Factoring
Guided Practice
Example 2
Solve 8x2 – 8 = –x2 + 56 by factoring.
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5.2.2: Factoring
Guided Practice: Example 2, continued
1. Rewrite the equation so that all terms are
on one side.
8x2 – 8 = –x2 + 56
Original equation
9x2 – 8 = 56
9x2
– 64 = 0
Subtract 56 from
both sides.
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5.2.2: Factoring
Guided Practice: Example 2, continued
2. Factor the difference of squares.
The expression on the left side can be rewritten in the
form (3x)2 – 82.
We can use this form to rewrite the expression as the
difference of squares to factor the expression.
(3x + 8)(3x – 8) = 0
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5.2.2: Factoring
Guided Practice: Example 2, continued
3. Use the Zero Product Property to solve.
The expression will equal 0 only when one of the
factors is equal to 0.
Set each factor equal to 0 and solve.
3x + 8 = 0
3x = -8
8
x=3
3x - 8 = 0
3x = 8
8
x=
3
8
8
8x2 – 8 = –x2 + 56 when x = - or .
3
3
5.2.2: Factoring
✔
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Guided Practice: Example 2, continued
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5.2.2: Factoring
Guided Practice
Example 3
Solve x2 + 8x = 20 by factoring.
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5.2.2: Factoring
Guided Practice: Example 3, continued
1. Rewrite the equation so that all terms are
on one side of the equation.
x2 + 8x = 20
Original equation
x2 + 8x – 20 = 0
Subtract 20 from
both sides.
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5.2.2: Factoring
Guided Practice: Example 3, continued
2. Find the factors.
The lead coefficient of the expression is 1, so begin
by finding two numbers whose product is –20 and
whose sum is 8.
The numbers are –2 and 10 because (–2)(10) = –20
and –2 + 10 = 8.
Therefore, the factors are (x – 2) and (x + 10).
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5.2.2: Factoring
Guided Practice: Example 3, continued
3. Write the expression as the product of its
factors.
(x – 2)(x + 10) = 0
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5.2.2: Factoring
Guided Practice: Example 3, continued
4. Use the Zero Product Property to solve.
The expression will equal 0 only when one of the
factors is equal to 0. Set each factor equal to 0 and
solve.
x–2=0
x + 10 = 0
x=2
x = –10
x2 + 8x = 20 when x = 2 or x = –10.
✔
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5.2.2: Factoring
Guided Practice: Example 3, continued
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5.2.2: Factoring