Solving Quadratic

1
Chapter 4 Quadratic and Polynomial Equations
4.2
Solving Quadratic Equations by Factoring
4.2.1
MATHPOWERTM 11, WESTERN EDITION
2
Factoring Simple Trinomials
Simple Trinomials The coefficient of the x2
term is 1.
The last term is the product of the constant
terms.
Recall (x 6)(x 4) x2 10x 24
The middle term is the sum of the constant terms
of the binomial.
Therefore, to factor simple trinomials use the
idea of the sum/product.
Factor
x2 9x 20
x2 13x 36
x2 - 11x 24
x2 x - 20
(x 5)(x 4)
(x 9)(x 4)
(x 5)(x - 4)
(x - 8)(x - 3)
4.2.2
3
Factoring Simple Trinomials contd
Keep in mind, that there are three possibilities

• If the last term is positive and the middle term
  positive,
• then both terms in the answer are positive.

x2 5x 6
x2 12x 32
(x 3)(x 2)
(x 8)(x 4)
2. If the last term is positive and the middle
term negative, then both terms in the answer
are negative.
x2 - 15x 56
x2 - 9x 8
(x - 7)(x - 8)
(x - 8)(x - 1)
3. If the last term is negative, then one answer
is positive and the other is negative.
x2 - 7x - 18
x2 5x - 36
(x - 4)(x 9)
(x - 9)(x 2)
4.2.3
4
Factoring General Trinomials
Recall (3x 2)(x 5)
3x2 15x 2x 10 3x2 17x
10
Note that the sum of 15x 2x is the middle term,
17x.
The product of 15x x 2x is 30x2. This is the same
as the product of the first and last terms of
the trinomial, 3x2 x 10 30x2.
Therefore, to factor 3x2 17x 10, look for two
numbers that have a product of 30 and a sum of
17.
4.2.4
5
Factoring General Trinomials - the Decomposition
Method
The numbers are -6 and -4 because -6 x -4
24 -6 -4 -10.
The product is 3 x 8 24. The sum is -10.
3x2 - 10x 8
Rewrite the middle term of the polynomial using
-6 and -4. (-6x - 4x is just another way
of expressing -10x.)
3x2 - 6x - 4x 8
- 4(x - 2)
3x(x - 2)
Factor by grouping.
(x - 2)(3x - 4)
4.2.5
6
Factoring General Trinomials - Some Examples
product is - 90 sum is 1
1. 6x2 x - 15
10 -9
6x2 10x - 9x - 15
2x(3x 5)
Check using foil (3x 5)(2x - 3)
6x2 - 9x 10x 15
6x2 x - 15
- 3(3x 5)
(3x 5)(2x - 3)
product is -36 sum is -5
2. 12y2 - 5y - 3
4 -9
12y2 - 9y 4y - 3
3y(4y - 3)
1(4y - 3)
(4y - 3)(3y 1)
3. 3x2 20x 12
product is 36 sum is 20
2 18
3x2 18x 2x 12
3x(x 6)
2(x 6)
(x 6)(3x 2)
4.2.6
7
Factoring General Trinomials - the Inspection
Method
1. List the pairs of factors of the first
and last terms.
Factor 6x2 - 11x 3
1 x 1 1 6 x 3 18 Sum 19
2 x 1 2 3 x 3 9 Sum 11
1 x 3 3 6 x 1 6 Sum 9
2. Do a cross product sum to find the sum
equal to the middle term.
1 6 2 3
1 3
3. Place the terms in the brackets,
keeping in mind FOIL. (place in opposite
brackets.)
Not equal to the middle term so dont use. Try
again.
This sum is equal to the middle term, so use
these numbers.
Not equal to the middle term so dont use. Try
again.
(2x - 3)(3x - 1)
Check using foil 6x2 - 2x - 9x 3 6x2 -
11x 3
4.2.7
8
Solving Quadratic Equations
To solve quadratic equations by factoring, apply
the Zero Product Property which states that, if
the product of two real numbers is zero, then
one or both of the numbers must be zero. Thus, if
ab 0, then a 0, or b 0, or a 0 and b
0.
Solve the following
1 1 3 14 2
7
x2 - 10x 16 0 (x - 8)(x - 2) 0 x - 8 0
or x - 2 0 x 8 or x 2
(3x - 2)(x 7) 0 3x - 2 0 or x 7 0
3x2 19x - 14 0
- 2 21 19
x -7
or
x2 8x 0 x(x 8) 0 x 0 or x 8 0 x
0 or x -8
4.2.8
9
Writing a Quadratic Equation With Given Roots
Find the equation, given the following roots
a) x -6, 3
Therefore, x -6 and x 3. The factors are (x
6) and (x - 3). (x 6)(x - 3) 0 x2 3x - 18
0
b)
Thus, 2x -3 and 3x 4. The factors are (2x
3) and (3x - 4). (2x 3)(3x - 4) 0 6x2
x - 12 0
4.2.9
10
Assignment
Suggested Questions
Pages 160-162 1, 5, 7, 15, 18, 21, 23, 25, 27,
29, 31, 33, 35, 39, 41, 49, 53, 55, 57, 59, 61,
63, 65, 70, 72, 75, 89
4.2.10