Notes on Solving Quadratic Equations #1
Name_________________________
Suppose three quantities (x, y, and z), when multiplied, have a product of zero. In other words,
x y z  0.
If this is true, what else must be true about x, y, and z?__________________________________
If two or more quantities have a product of zero, then one or more of the quantities must
___________________________.
--------------------------------------------------------------------------------------------------------------------This property, known as the Zero-Product Property, can be combined with Factoring in order
to solve equations.
1.
2.
3.
4.
5.
Solving Quadratic Equations by Factoring
Collect all terms on one side of the equation using addition/subtraction. The other side
should be zero. You only can combine like terms.
Rewrite the side with the terms in standard form.
Factor that side.
Create new equations by setting individual factors equal to zero.
Solve the new equations.
Example #1
Solve.
3x 2  6 x  1  2 x 2  9
1.
Subtract 2x 2 from both sides and add 9 in order to
get all terms on the same side.
x2  6x  8  0
2.
Make sure it is in standard form.
( x  4)( x  2)  0
3.
Factor.
x40
x20
4.
Set each factor equal to zero.
x  4
x  2
5.
Solve the equations.
The solutions to the equation are x = -4 and x = -2.
--------------------------------------------------------------------------------------------------------------------Solve: x 2  7 x  24  2 x
Example #2
Solve.
9 x 2  10 x  16  10 x 2
1.
Subtract 10x 2 from both sides in order to get all
of the terms on the same side.
 x 2  10 x  16  0
2.
We have a slight problem that will sometimes
occur. In order to factor the expression, we need
a = 1, not -1. In order to fix the problem, let's
multiply both sides by -1. The result --- every sign
in the equation gets changed.
( x  8)( x  2)  0
3.
Factor.
x 8  0
4.
Set each factor equal to zero.
x 2  10 x  16  0
x 8
x20
x2
5.
Solve the equations.
--------------------------------------------------------------------------------------------------------------------Solve: 4 x 2  6 x  5  5 x 2
--------------------------------------------------------------------------------------------------------------------Example #3
Solve.
4 x  7  x2  0
1.
Fortunately, all of the terms are already on the left
side - this step can be skipped.
x2  4x  7  0
2.
Rewrite the left side in standard form.
***UH OH!!!***
3.
Regretfully, the left side cannot be factored. Thus,
we will state, "Cannot solve by factoring."
--------------------------------------------------------------------------------------------------------------------Checking Your Answers
A great part about this section is that a person can check his/her answers. All that one has to do
is plug the solutions into the original equation, and then see if it produces true statements.
Let's check the answers for Example 2 in the space below.
Homework on Solving Quadratic Equations #1
Name_________________________
Solve the equations for x.
1.
11x  x 2  30  0
2.
x 2  6 x  24  x
3.
8 x 2  10 x  52  7 x 2  12 x  4
4.
x 2  x  40  2 x  2
5.
x2  5x  6  0
6.
3x 2  19 x  48  4 x 2
7.
x 2  2 x  14  3
8.
3x  x 2  1  1
9.
0   x 2  9 x  22
10.
2 x 2  5 x  48  3x 2  9 x  3
--------------------------------------------------------------------------------------------------------------------1. x  6, x  5
2. x  8, x  3
3. x  8, x  6
4. x  7, x  6
5. x  6, x  1
6. x  3, x  16
7. can't be solved by factoring
8. x  2, x  1
9. x  11, x  2
10. x  9, x  5