Download Common Algebra Errors

Common Algebra Errors
Division by zero
Your algebra 1 teacher told you this is wrong. Your geometry
teacher told you this is wrong. Your algebra 2 teacher told you
this is wrong. Put 2/0 in your calculator and even IT will tell
you it’s wrong!
Embarrassing isn’t it?
Everyone knows that 0/2 = 0: nothing out of 2 is still nothing.
Still, far too many people will claim that 2/0 = 0 or even that
2/0 = 2!
You might not even not know you’re dividing by 0, as you’ll
see in our next example!
a=b
Given.
ab = a2
Multiply both sides by a,
ab – b2 = a2 – b2
Subtract b2 from both sides.
b(a – b) = (a+b)(a-b) Factor both sides.
b=a+b
Divide both sides by a – b
b = 2b
We assumed a = b above.
1=2
Divide both sides by b.
What happened? Dividing by a – b is dividing by 0, since a = b.
Missed Parentheses
Example 1: Square 4x.
Incorrect: 4x2
Correct: (4x) 2 = (4) 2(x)2 = 16x2
Example 2: Square –3
Incorrect: -32 = -(3)(3) = -9
Correct: (-3)2 = (-3)(-3) = 9
Missed Parentheses
Example 3: Subtract 4x – 5 from x2 + 3x - 5
Incorrect: x2 + 3x – 5 – 4x – 5 = x2 – x – 10
Correct: x2 + 3x – 5 – (4x – 5) = x2 + 3x – 5 – 4x + 5 = x2 - x
Example 4: Convert
Incorrect: = 7x1/2
Correct: = (7x)1/2
7 x to fractional exponents.
Incomplete Distribution
Example 1: Multiply 4(2x2 – 10)
Incorrect: = 8x2 – 10
Correct: 8x2 - 40
Example 2: Multiply 3(2x – 5)2
Incorrect: = (6x – 15)2 = 36x2 – 180x + 225
Correct: = 3(4x2 – 20x + 25) = 12x2 – 60x +75
Remember that you must expand the exponent before
distributing!
Assumed Addition
x  y 
2
x y
2
2
x y  x  y
1
1 1
 
x y x y
You can prove that these aren’t true by simply plugging in some
numbers…try it!
Canceling Errors
Example 1: Simplify 3x3 – x
x
Incorrect: 3x2 – x
Correct:
OR 3x3 – 1
x(3x 2  1)

 3x 2  1
x
We can’t cancel the x in the denominator against only one of
the x’s in the numerator.
Canceling Errors…continued
Likewise, we can’t simplify 3x3 – 1
x
Can you tell why?
Canceling Errors…one more
Solve 2x2 = x.
We can cancel one x from both sides to get 2x = 1 or x = ½.
This is a solution, but there is something we missed!
Can you tell what it is?
Canceling Errors…one more
Here’s a better way to solve 2x2 = x.
First we get everything on one side: 2x2 – x =0
Next, we factor: x(2x – 1) = 0
From this we see that x = 0 OR 2x – 1 = 0 will work.
We still get x = ½, but we also have x = 0 as a solution.
What’s my square root?
Remember that square roots are positive or zero.
16  4 NOT
+
4
However, when solving x2 = 16, the answer is x = + 4.
WHY?
What’s my square root?
continued
Here’s the solution with all steps included:
x 2  16
x   16
x  4
NOTICE: The + shows up BEFORE not after we take
the square root.
Fraction Frenzy
We can write
2 or
x
3
2 .
3x
But when we write
2/3 x which do we mean? Writing
2
3
x is also confusing.
Also, writing a + b as a + b/c + d might be interpreted as
c+d
a + b + d. To make it clear, write (a + b)/(c + d).
c
Exercises
1. Square 6x.
2. Square –3.
3. Multiply 5(x – 4)2
4. Expand (x + y)2.
5. Solve 3x2 = x.
6. Does 3a  6b  3a  6b ?
7. Add
1 1

x y
.
8. Can we simplify x2 – 1 ?
x2