7.6 Factoring: A General Review
7.7 Solving Quadratic Equations by
Factoring
Checklist for factoring polynomials of any type.
• 1. If the polynomial has a greatest common factor other than 1,
factor out this GCF
• 2. If the polynomial has two terms (binomial), check to see if it is a
difference of two squares or sum/difference of two cubes and factor
accordingly, if it is a sum of squares it will not factor
• 3. If the polynomial has three terms (trinomial), then either it is a
perfect square trinomial, which will factor into a binomial squared, or
it is not a perfect square trinomial, in which you can try to factor
using methods developed in 7.2 and 7.3
• 4. If the polynomial has more than three terms, try to factor by
grouping
• 5. As a final check, see if any of the factors you have written can be
factored again or further. If you have overlooked a common factor
you can catch it here. Remember, some polynomials cannot be
factored, in which case they are considered prime.
Practice
3 x  27 x
4
3
2 x  20 x  50 x
5
4
y  36 y
5
3
3
15a  a  2
2
4 x  12 x  40 x
4
3
2
3ab  9a  2b  6
7.6 1-74 odd
# 68. ( x  5)  2( x  5)
3
2
# 74. (3 x  1)  (3 x  1)
3
# 66.  3 y  81 y
4
#58. 16 x  16 x  1
2
7.7 Solving Quadratic Equations by
Factoring
This will be the exact same thing that we have been doing, except
Now there will be an equals sign and another expression on the other
side, the overall goal is to make one side of the equal sign a zero and
then factor the remaining side. Then use what we call the ZEROFACTOR PROPERTY
Definition: Quadratic Equations
• Any equation that can be put in the form
ax2+bx+c=0, where a, b, and c are real numbers
(a cannot equal 0), is called a quadratic equation.
The equation ax2+bx+c=0 is called standard form
for a quadratic equation.
Zero-Factor Property
• Let a and b represent real numbers. If a x b =0,
then a=0 or b=0.
• Thus if we factor x2+5x+6=0, we are left with
(x+2)(x+3)=0. Because there is multiplication and
our product is zero, one or both factors, must
indeed be zero themselves, thus we make the
equations x+2=0 and x+3=0 and solve for x. This
will then provide the only values for x that will
force the quadratic equation to equate to zero.
Solve the quadratic equation
2 x  5x  3
2
Solve
16a  25  0
2
What to do when a GCF is factored out
and the GCF is just a constant.
2 x  10 x  12
2
What to do when the GCF has a
variable term.
5 x  2 x
2
• On occasions you may have to multiply
something out and combine like terms before
you can factor. (initially the problem may
actually look pre-factored but it is not because
the other side is not a zero).
x(2 x  3)  44
Example
x(13  x)  40
Harder example
( x  2)  ( x  1)  x
2
2
2
Solve: this example is not a quadratic,
(we call it a cubic) but it can be solved
in the same manner
4 x  x  5x
3
2
Section 7.7 1-84 odd.