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Lesson 8-3 Factoring Trinomials (Quadratic Equations x2+bx+c)
Factoring x2+bx+c – find two integers m and p with a sum of b and a product of c, then write
x2+bx+c as (x+m)(x+p).
Example 1 b and c are Positive
Factor x2 + 16x + 63.
In this trinomial, b = 16 and c = 63. Since c is positive and b is positive, you need to find two
positive factors with a sum of 16 and a product of 63. Make an organized list of the factors of 63,
and look for the pair of factors with a sum of 16.
Factors of 63
1, 63
3, 21
7, 9
Sum of Factors
64
24
16
x2 + 16x + 63 = (x + m)(x + p)
= (x + 7)(x + 9)
Check
The correct factors are 7 and 9.
Write the pattern.
m = 7 and p = 9
You can check this result by multiplying the two factors. The product should be equal
to the original equation.
(x + 7)(x + 9) = x2 + 9x + 7x + 63
FOIL method
= x2 + 16x + 63 
Simplify.
Example 2 b is Negative and c is Positive
Factor x2 – 11x + 24.
In this trinomial, b = –11 and c = 24. Since c is positive and b is negative, you need to find two
negative factors with a sum of –11 and a product of 24. Make a list of the negative factors of 24,
and look for the pair of factors with the sum of –11.
Factors of 24
–1, –24
–2, –12
–3, –8
–4, –6
Sum of Factors
–25
–14
–11
The correct factors are –3 and –8.
–10
x2 – 11x + 24 = (x + m)(x + p)
= (x – 3)(x – 8)
Check
Write the pattern.
m = –3 and p = –8
You can check this result by using a
graphing calculator. Graph y = x2 – 11x + 24
and y = (x – 3)(x – 8) on the same screen.
Since only one graph appears, the two
graphs must coincide. Therefore, the
trinomial has been factored correctly. 
Example 3 c is Negative
Factor each polynomial.
a. x2 + x – 6
In this trinomial, b = 1 and c = -6. Since c is negative, the factors m and p have opposite
signs. So either m or p is negative, but not both. Since b is positive, the factor with the
greater absolute value is also positive.
List the factors of –6, where one factor of each pair is negative. Look for the pair of
factors with a sum of 1.
Factors of –15
Sum of Factors
–1, 6
5
1, –6
–5
–2, 3
1
The correct factors are –2 and 3.
2, –3
–1
x2 + x – 6 = (x + m)(x + p)
= (x – 2)(x + 3)
Write the pattern.
m = –2 and p = 3
b. x2 – 4x – 21
In this trinomial, b = -4 and c = -21. Either m or p is negative, but not both. Since b is
negative, the factor with the greater absolute value is also negative.
List the factors of 21, where one factor of each pair is negative. Look for the pair of factors
with a sum of 4.
Factors of –21
Sum of Factors
–1, 21
20
1, –21
–20
–3, 7
4
3, –7
–4
The correct factors are 3 and –7.
x2 – 4x – 21 = (x + m)(x + p)
= (x + 3)(x – 7)
CHECK
Write the pattern.
m = 3 and p = –7
Graph y = x2 – 4x – 21 and y = (x + 3)(x  7) on the same screen. Since the graphs
coincide, the trinomial has been factored correctly. 
Quadratic Equation – an equation of the form x2+bx+c =0. To solve: factor the quadratic and set the
factors equal to zero.
Example 4 Solve an Equation by Factoring
Solve x2 – 8x + 7 = 0. Check your solutions.
x2 – 8x + 7 = 0
(x – 1)(x – 7) = 0
x–1=0
or x – 7 = 0
x=1
x=7
Original equation
Factor.
Zero Product Property
Solve each equation.
The roots are 1 and 7.
CHECK Substitute 1 and 7 for x in the original equation.
x2 – 8x + 7 = 0
x2 – 8x + 7 = 0
?
(1)2 – 8(1) + 7  0
?
(7)2 – 8(7) + 7  0
?
1–8+7  0
0=0
?
49  56 + 7  0
0=0
Real-World Example 5 Solve a Problem by Factoring
NUMBER THEORY Find two consecutive integers whose product is 56.
Understand You are to find two consecutive integers that have a product of 56.
Plan
Write an equation representing the product of the two integers.
Solve
Let x = the first integer. Then x + 1 = the next consecutive integer.
first integer
x
times

x(x + 1) = 56
x2 + x = 56
2
x + x – 56 = 0
(x + 8)(x – 7) = 0
x + 8 = 0 or x – 7 = 0
x = –8
x=7
consecutive integer
equals
56
(x + 1)
=
56
Write the equation.
Multiply.
Subtract 56 from each side.
Factor.
Zero Product Property
Solve each equation.
One possible set of consecutive integers is –8 and –8 + 1 or –7. The
other is 7 and 7 + 1 or 8.
Check
–8  (–7) = 56 and 7  8 = 56, so the solution checks.