Brain Teasers: Equation Gone Wrong

Hey all! Can anyone figure out what went wrong?

a = b

ab = b^2

ab - a^2 = b^2 - a^2

a(b - a) = (b - a)(b + a)

a = b + a

                                                          a = b;          a = a + a

a = 2a

1 = 2?

What happened?

Solution: Look at Steps 4 and 5 in my justification above:

a(b - a) = (b - a)(b + a)

a = b + a

If a = b, then b - a = 0, so the real justification would be:

a(0) = (0)(b + a)

0 = 0

This means I divided by 0 in my justification, which is impossible according to the laws of mathematics.

Hope you enjoyed!

Fast Facts

Hey! Here are some fun math tips.


Fact 1:

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

See? The sum of odd consecutive numbers starting from 0, will always be a perfect square number no matter how far you go. Check it out!

Fact 2:


4% of 100 = 100 % of 4
6% of 38 = 38% of 6
5% of 10 = 10% of 5
7% of 50 = 50% of 7

Catch the drift?

Fact 3:

(a * b) * (c * d) = 100(ac) + 10(ad + bc) + (bd)

This is just the Rainbow Method simplified into an equation!

Fact 4: 

To square 2 digit numbers ending with ‘5’ (Example: 75 × 75)
1.  Take the first digit ‘7’ multiply by the number after ‘7’ => 7 × 8 = 56 

Now add 25 after the number. 56(25) = 5625

Hope you all enjoyed!

Factoring Quadratic Expressions: Box Method

Hello all. Sorry for not posting a blog in a while. I've been having a lot on my hands and I couldn't find the time to get to this. Now I do :). Today, I will be teaching you all a technique on how to factor quadratic expressions in the form ax^2 + bx + c.

The example we will be using is: 2x^2 - 5x + 3.

To factor this expression, we will use the Box Method. First, create a box with 5 squares.

Step 1: Insert the first term in the 2nd square and the last term in the 5th square. Then multiply those terms together, and put the product to the side of the box as shown:

Step 2: Now find two terms that multiply together to get the product on the right, BUT ALSO, add up together to get the middle term in the original expression. In this case, we need to find two terms that if multiplied, get us 6x^2, but also if added together, get us -5x. Put each term on boxes 3 and 4. It doesn't matter which term goes in which box.

Step 5: Now its time to calculate the factorization of this equation. First, calculate the GCF of the two terms of the top 2 boxes and put the GCF to the left of Box 2. Then do the same thing with the bottom 2 boxes and put the GCF to the left of Box 4. (I made a typo: the GCF next to Box 4 is -3, and not 3).

Step 6: Now you have to calculate the GCF of the two terms of the left 2 boxes and put the GCF under Box 4. Then do the same thing with the right 2 boxes and put the GCF under Box 5. 

Step 7: Now we know the numbers of our factoring. Just put parentheses around the GCFs to the left of the square to get the first "term" in the factoring, and put parentheses around the GCFs under the square to the get the second "term" in the factoring.

Therefore, if we factor 2x^2 - 5x + 3, we get: (2x - 3)(x - 1). 

Thanks for reading this method on factoring! If you have any questions, please post comments down below.