Cubing - Two Digit Number

Category: Arithmetic, Algebra

"Published in Suisun City, California, USA"

How do you cube a two digit number without using a calculator in a shorter way or faster? Is that possible? Yes, it is. I will show you the technique. Let's consider this one:

(23)³

By usual method,

           2 3
    x     2 3
  ---------------
        1  6 9
        4 6
  ---------------
            1
            2
        5 2 9
   x      2 3
  ---------------
     1 5 8 7
  1 0 5 8
  ---------------
  1 2 1 6 7

Now, let's consider the better way. Cube each digits first and write this way:

Next, we need the next two terms at the middle. Let's consider the next steps. 

Square the first digit and then multiply with the second digit and then multiply by 3. In this case, 3(2)²(3) = 36.

Square the second digit and then multiply with the first digit and then multiply by 3. In this case, 3(2)(3)² = 54.

Therefore, the answer is 12167.

Finding Unknown Coefficients - Quadratic Equation

Category: Algebra

"Published in Newark, California, USA"

Find the value of k to the given equation

if one root is double the other.

Solution:

Consider the given equation

If a = 2, b = k, and c = 25, then substitute these values to the Sum of the Roots Formula

From the word problem, if r₁ = 2r₂, then

From the Product of the Roots Formula

Substitute the value of the variables to the above equation

Therefore, the answer is k = 15. 

Squaring, Two - Three Digit Number

Category: Arithmetic, Algebra

"Published in Newark, California, USA"

How do you square a two or a three digit number without using a calculator in a shorter way and faster? Well, let's consider first the squaring of a two digit number. Let's have this one:

(32)²

By usual method,          
          
          3  2
     x   3  2
-------------------       
       1   6 4
       9  6    
-------------------
    1 0  2  4

Now, let's consider the better way:

Next, let's get the middle number:

Therefore, the answer is 1024.

There's another way for squaring a two digit number that ends with 5. Let's consider this one:

(65)²

Here's the technique, square the last digit, which is 5 as follows:

                              ____25

Next, multiply the next digit by its consecutive number. In this case, 7 is the next number to 6. Therefore, 6 x 7 = 42. Your final answer is: 4225

Now, let's consider the squaring of a three digit number. let's have this one:

(123)²

By usual method,

             1  2  3
       x    1  2  3
    --------------------
             1
         1   3  6  9
         2  4  6
     1  2  3
   ---------------------
     1  5  1  2  9


Now, let's consider the better way:

Next, let's get the other numbers:

Add all the numbers in each lines:

Therefore, the answer is 15129.