Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 1
Shortcuts to Factors and Zeros
In the past two lessons we covered a general strategy for factoring ax2 + bx + c,
finding zeros of ax2 + bx + c, and solving equations involving quadratics.
While that method will work for any quadratic that can be factored, there are some
special situations where an easier method is available.
The case where c=0: factoring out a common factor
For functions that have the form f(x) = ax2 + bx, the “common factor” x can always be
“factored out” from both terms. That is, one of the two factors is always x, and the other factor
can be found by removing an x from each of the terms.
Example: Factor x2 – 16x.
x2 – 16x = x(x – 16)
Example: Factor 2x2 + 7x.
2x2 + 7x = x(2x + 7)
Note that in this f(x) = ax2 + bx case, since x is always a factor, the number 0 is always a zero.
This fact can be also be established by evaluating: f(0) = a · 02 + b · 0 = 0.
Finding zeros or solutions when b=0: using a square root
If you’re asked to find a zero or a solution when the bx term is missing, here’s a method that may
be easier than factoring: get the term with the x2 on one side of the equation, a number on the
other side of the equation, and then use square root. You’ll also need to remember that if x2 is
known to equal a certain number, there are two possibilities for x, positive and negative.
In the examples below, note the use of the symbol ± to show the two solutions.
For instance, ±4 stands for two numbers: 4 and –4.
Example: Solve x2 = 16.
Solution: x = ±4
Example: Solve x2 = 10.
Solution: x = ± 10 ≈ ±3.162
Example: Find the zeros of f(x) = x2 – 10.
x2 – 10 = 0
x2
= 10
x
= ± 10 ≈ ±3.162
Factoring when b=0: difference of squares
Example: Factor 4x2 – 9.
4x2 – 9 = (2x)2 – 32 = (2x + 3)(2x – 3)
Example: Factor x2 – 10.
x2 – 10 = x2 – ( 10 )2 = (x – 10 )(x + 10 )
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 2
Problem Set
1. Do these problems using the “common factor” method. It should not be necessary to write
any multiplication tables.
a. Factor 3x2 – 8x.
b. Find the zeros of f(x) = x2 + 6x.
c. Find the zeros of f(x) = 5x2 – 8x.
d. Solve the equation x2 = 10x.
Hint: First convert to a “find the zeros” problem.
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 3
2. Find solutions and zeros as requested below. All of these problems fall into the b=0 case,
so can be done with the help of square roots. Every part has two answers; make sure you find
both of them.
a. Solve x2 = 81.
b. Solve 3x2 = 108.
Hint: Divide both sides by 3, then use same method as a.
c. Solve x2 = 108.
Hint: You’ll need to use your calculator to get the values.
d. Find the zeros of x2 – 25.
e. Find the zeros of x2 – 26.
f. Solve x2 – 49 = 0.
g. Find the zeros of 2x2 – 18.
Hint: First steps: write 2x2 – 18 = 0 then divide by 2.
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 4
3. Factor these functions. All of these functions have b = 0, so the “difference of squares”
approach can be used. It should not be necessary to write any multiplication tables.
a. Factor f(x) = x2 – 4.
b. Factor f(x) = x2 – 5.
c. Factor f(x) = 9x2 – 4.
Hint: 9x2 is equal to (3x)2.
d. Factor f(x) = 49x2 – 1.
e. Factor f(x) = 100x2 – 100.
f. Factor f(x) = 3x2 – 48.
f(x)= 3(x2 – 16)
g. Factor f(x) = 5x2 – 80.
Hint: First step shown. Then factor the (x2 – 16).
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 5
4. Find the zeros of function f(x) = 16x2 – 4 two different ways. You should get the same
answers from both methods.
a. by writing 16x2 – 4 = 0 and solving the equation using a square root.
b. by factoring 16x2 – 4 then using the factors to find the zeros.
5. Find the zeros of function f(x) = 50x2 – 2 two different ways. You should get the same
answers from both methods.
a. by writing 50x2 – 2 = 0 and solving the equation using a square root.
Need a hint? see problem 2g
b. by factoring 50x2 – 2 then using the factors to find the zeros.
Need a hint? see problem 3f
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 6
6. When you solved equations in problem 3, there were always two solutions. In the following
set of problems, that’s not always true.
a. Solve x2 = 16.
b. Solve x2 = –16.
c. Solve x2 = 19.
d. Solve x2 = –19.
e. Solve x2 = 0.
7. Look at all the equations you’ve solved today where x2 = [number].
Sometimes there are two solutions, sometimes there’s just one, and sometimes there are none.
Based on the number in the equation, how can you tell how many solutions there will be?
Name:
Date: _______________
Algebra 2
Shortcuts to Factors and Zeros page 7
ANSWERS to Shortcuts to Factors and Zeros:
1.
2.
a.
b.
c.
d.
a.
x 3x  8
x = 0, x = -6
x = 0, x = 8/5
x = 0, x = 10
x  9
b. x  6
c.
x   108  10.392
d. x  5
e.
x   26  5.1
f.
x  7
g. x  3
3.
a. f x  x  2x  2



b.
f x   x  5 x  5
c.
f x  3x  23x  2
d.
f x  7 x  17 x  1
e.
f x  10x  1010x  10  100x  1x  1
f.
f x  3x  4x  4
g.
f x  5x  4x  4
4. a. x  
5. a. and b.
1
2
b.
x
f x  4x  24x  2 = 0, so same answer as a.
1
5
6. a. x  4 b. No real solution
c. x   19
d. No real solution e. x = 0
7. If number >0 there are two solutions. If number < 0 there are no real solutions. If number
= 0, there is one solution.