### 6AM Quiz (20110730) What's Wrong with the Area of Square Sticker

Write down the expression for the area of the square (not drawn to scale) shown.
Expand and simplify the expression.

Rachel's solution:
Area of square = length x length
= ( x + 5 ) 2
= x2 + 52
= x2 + 25 cm2

There is something wrong with Rachel's solution.
• Show how you would solve the problem.
• Explain the error in Rachel's solution.

1. (x+5)^2=(x+5)(x+5)
=x^2+5x+5x+25
=x^2+10x+25
She squared the x and the 5 itself instead of squaring the whole (x+5)

2. I would solve the problem like this:

(x + 5)²
= (x + 5)(x + 5)
= x² + 5x + 5x + 25
= x² + 10x + 25 (Answer)

What was her error:

She put ( x + 5 )² = x² + 5², whereas, it should have been
( x + 5 )² = (x + 5)(x + 5). This is because (x + 5) is itself one whole term, so you have to square the whole thing and not only x and 5.

Here is an example:
Lets take x as 2.
(2 + 5)²
= 7²
2² + 5²
= 4 + 25
Thus, (2 + 5)² is never equal to 2² + 5², unless x is equal to 0, which is not the case here as a rectangle cannot have a length of 0 cm.

3. To solve the problem.
(x+5)^2 = (x+5)(x+5)
= (x^2)+2(5x)+(5^2)
= x^2+10x+25

Rachel's mistake:
(x+5)^2 = (x^2+5^2)

She is using the distributive law wrongly.
If used distributive law, it should be (x+5)^2
= (x[x+5]+5[x+5])
This would only be correct if x is >0 .

Also by substitution,
Let x be 3.
(3+5)^2
= 8^2
= 64

3^2+5^2
=9+25
=34

Owen here.

4. This comment has been removed by the author.

5. (x+5)^2=(x+5)(x+5)
= x^2+10x+25

Instead of calculating the entire area for the square, she missed two pieces of areas which has 5x each. see below.
x + 5
---------------------
|..............|.........|
|...x^2.....|..5x...|
|..............|.........|
|..............|.........|
---------------------
|..............|.........|
|....5x......|..25...|
|..............|.........|
---------------------

6. I would solve it like this :
(x+5)^2 = (x+5)(x+5)
= x^2 + 5x + 5x + 25
= x^2 + 10x + 25 cm

Rachel should not have conveniently squared the amount in the brackets as it would not give her the accurate answer .

7. My solution:
(x+5)^2 = (x+5)(x+5)
= x^2+5x+5x+25
= x^2+10x+25

Rachel is wrong as by just squaring the numbers in the brackets, it will not be accurate. By using a square as example, to find its area, we have to take the length X breadth, and since the length and breadth is the same, it will be for e.g. (x+5)^2 which is (x+5) x (x+5)

(x+5)^2 = (x+5)(x+5) NOT x^2+5^2
= x^2+5x+5x+25
== x^2+10x+25

Rachel squared the amounts in the brackets rather than squaring the numbers in the brackets as a whole.

9. Rachel just simply squared the equation in the bracket to get x^2+5^2=x^2+25. But the formal for (a+b)^2 is a^2 + 2ab + b^2.

So
Area of square = length x length
= ( x + 5 ) 2
= ( x + 5 ) ( x + 5 )
= x * x + x*5 + 5*x + 5 * 5
= x^2 + 5x + 5x + 25
= X^2 + 10 x + 25

10. Rachel just squared the terms but not the whole sum.

The correct method should be using the foil method to do
(x+5)(x+5)
=x^2+10x+25

11. My solution to the question :

Area of square=length*length
=(x+5)^2
=(x+5)(x+5) Rewrite the expression
=x^2+5x+5x+25 Use FOIL method .
=x^2+10x+25 Simplify

Ans:x^2+10x+25cm^2

Rachel should have used the prove or FOIL method to expand the expression but Rachel just expand the two terms in the brackets.(x+5)^2 is length*length so the expression should be rewrite as before multipllying (x+5)(x+5).

12. How I would solve.
(x+5)^2=(x+5)(x+5)
using the foil method, I will get x^2+5x+5x+25=x^2+10x+25

Rachel's error is that she did not make (x+5)^2 into(x+5)(x+5) therefore, she could not use the foil method.

13. (x+5)^2 = (x+5)(x+5)
= x^2+10x+25

She just squared them individually, thus she got it wrong :D

14. Rachel's solution:
Area of square = length x length
= (x+5)^2
= x^2 + 5^2
= x^2 + 25 cm^2

Firstly, when you square a term in brackets, you square the whole thing as one term, and not square it into the bracket as separate terms. (x+5)^2 is actually (x+5)(x+5) and not x^2+5^2.

My solution:
(x+5)^2
=(x+5)(x+5)
=x^2+25+10x

And my personal opinion is that the cm^2 was abit too abrupt and she should only add the unit at the end of the working to avoid confusion.

15. How I would solve the problem:

(x + 5)^2
=(x + 5) (x + 5)
=x^2 + 10x + 25

What her error was:

She squared individuals (x and 5), when she was supposed to square the whole thing.

16. Chan Shawn Kit7/30/2011 01:22:00 PM

How would I answer the question: (x+5)(x+5)cm
= x^2 + 5x + 5x + 25 (FOIL method)
= x^2 + 10x + 25

Rachel's error:
She expanded (x+5)^2 into x^2 and 5^2, neglecting the 10x (An error i always made -.-).

(x+5)(x+5)
= x^2+5x+5x+25
=x^2+10x+25

Rachel's error:
She squared the expression (x+5) individually to x^2 and 5^2 instead of squaring (x+5) as a whole.

18. She have not follow the formula. If she did, her working will be like:
(x) ^ 2 + 2 (x) (5) + (5)^2=
x^2+10x+25

She missed out the 2(x)(5), thinking that removing the brackets and add in the power will be the answer and end up missing out that part and got the wrong answer.

19. I would solve it using the F.O.I.L. method:
(x+5)cm x (x+5)cm
= x^2+5x+5x+25
= x^2+10x+25

She did not understand the concept well and misunderstood that (x+5)^2 = x^2+5^2.

20. I would have solved it the same way as Rachel however I would use the F.O.I.L method.
Thus,
(x + 5)^2 = (x + 5) (x + 5)
= x^2 + 5x + 5x + 25
= x^2 + 10x + 25

When she squared the equation she only squared what was inside but not the entire.

21. ^^ Matthew here, sorry I was using my other acc.

22. I would have used the F.O.I.L method to solve the equation:
(x + 5)^2= (x + 5)(x + 5)
= x^2 + 5x + 5x + 25
= x^2 + 10x + 25

She squared each term individually into x^2 and 5^2, but should have squared the whole expression to (x+5)(x+5)

23. Well done, ladies and gentlemen. All of u are able to point out what ought yo be done :)

I'm also impressed by the variety of ways the explanation was done.
Keep up with the good work :)