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6.6 Quadratic Equations
Multiplying Binomials
• A binomial has 2 terms
• Examples: x + 3, 3x – 5, x2 + 2y2, a – 10b
• To multiply binomials use the FOIL method
First Last
2
x

x

x

4

3

x

3

4

x
 7 x  12
(x + 3)(x + 4) =
Inner
Outer
First Outer Inner Last
Examples:Multiply
• (x + 5)(x + 6)
• (3x – 4)( 5x + 3)
Common factors
• When factoring polynomials, first look for a
common factor in each term
• Example: The binomial below has the
factor 3 in each term
3x + 6y = (3)x + (3)2y = 3(x + 2y)
To factor the above polynomial we used the
distributive property.
ac + bc = c(a + b)
Examples: Factoring by
distributive property
1. 16n2 + 12n
2. 4x2 +20x -12
Examples: Difference of two squares
a2 –b2 = (a + b)(a – b)
1.
2.
3.
4.
5.
6.
x2 – 16
9y2 - 25
49x2 – 36 z2
x4 – 81
4x2y2 – b4
3x3 – 12x
Factoring Trinomials
• A trinomial has three terms.
• Example: x2 + 5x + 6
• If a trinomial factors, it factors into two
binomials.
Factoring trinomials
• First, look for a common factor.
• Then look for perfect square trinomials:
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
• If it is not a perfect square trinomial then
factor into two binomials.
Examples: Perfect square trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
1.
2.
3.
4.
5.
x2 + 10x + 25
x2 – 8y + 16
9x2 – 24x + 16
25x2 + 80xy + 64y2
8x2y – 24xy + 18y
Factoring trinomials of the form
x2 + bx + c
• x2 + bx + c = (x + ___)(x + ____)
• x2 - bx + c = (x - ___)(x - ____)
• To fill in the blanks look for factors of c
that add up to equal b
Examples: Factor. Check answers
by FOIL
•
•
•
•
x2-5x+6
x2+6x+8
x2-7x+10
x2+7x+12
Factoring trinomials of the form
x2 + bx - c
• x2 + bx - c = (x + ___)(x - ____)
Signs will be
different
• To fill in the blanks look for factors of c
that subtract to equal b
• If the 1st sign is negative place the larger
factor with the negative sign
• If the 1st sign is positive place the larger
factor with the positive sign
Examples: Factor
•
•
•
•
x2+2x-35
x2-4x-12
x2-2x+15
x2+5x-36
Factoring trinomials of the form
ax2 + bx + c
• One method is trial and error
• Try factors of a and c then FOIL to see if it
works
• Examples:
• 2x2 + 15x + 28 = (
+
)(
+ )
• 3x2 +7x – 20= (
+
)(
)
Alternative method: Factor by
grouping
• Factor by grouping is used to factor
polynomials with 4 terms
• Example: Factor 10x2 – 15x + 4x – 6
2 – 15x) + (4x – 6)
Group together
(10x
Factor out any
1st 2 terms and
5x(2x – 3) + 2(2x – 3) common
last 2 terms
factors in each
(2x – 3)(5x + 2)
group
Factor out (2x – 3)
from each term
Examples: Factor by grouping
Factoring trinomials using factor by
grouping.
• Since factor by grouping involves 4 terms
we want to rewrite the trinomial as a
polynomial with 4 terms
General trinomials: ax2 + bx + c
• Example: 2x2 + 5x – 3
• Multiply ac = 2(-3) = -6
Factors of –6
1, -6
2, -3
3, - 2
6, -1
• Select 6 and -1
Sum of factors
-5
-1
1
5*
Example cont’d
2x2 + (___ + ___) – 3
2x2 + (6x + -1x) – 3
(2x2 + 6x) + (-x – 3)
2x(x + 3) + -1(x + 3)
(x + 3)(2x – 1)
More examples
•
•
•
•
3x2 – 14x – 5
15m2 + 14m – 8
12x2 + 23p + 5
12 – 20x – 13x2
Zero Product Property
• To solve a quadratic equation by factoring
we will use the zero product property:
If ab = 0,
then a = 0 or b = 0
where a and b are any real numbers.
Examples: Solve by factoring
1. x2 – 3x – 28 = 0
2. x2 + 4x = 12
More examples
3. 16x2 = 49
4. 4x2 – 35x – 5 = 4
More examples
5. x2 + 6x = 13 = 4
6. 4x2 – 12x = 0
Examples: Solve by factoring
1. x2 – 3x – 28 = 0
2. x2 + 4x = 12
3. 4x2 – 35x – 5 = 4
4. 4x2 – 12x = 0
Examples: Solve by finding
square roots
1. 16x2 = 49
2. 5x2 – 180 = 0
3. 3x2 = 24
4. x2 – ¼ = 0
Quadratic formula
• To use the quadratic formula, the equation
must be in the form ax2 + bx + c = 0
b  b  4ac
x
2a
2
Examples: solve using the
quadratic formula
1. 2y2 + 4y = 30
2. x2 – 7x + 1 = 0
3. 5m2 + 7m = -3
4. x2 + 16 = 8x
Discriminant
• The discriminant is b2 – 4ac.
• The following chart describes the roots
based on the value of the discriminant.
b2 – 4ac
>0
<0
=0
Roots
Graph
2 real
Intersects x-axis twice
2 imaginary Does not intersect x-axis
1 real
Intersects x-axis once
Examine the determinants in the
previous examples to verify the
chart