X Squared

In algebra, when x is multiplied by itself, it is written as x². Here, x is the base and 2 is its exponent. In other words, for any variable x, x2 is an algebraic notation that refers to a number being multiplied by itself i.e., x . x=x². 
 

The value of x² can be found by multiplying the value of x with itself.
Let us take a few examples to see how:
Example 1. If x = 3
x² = 3 × 3 = 9

Example 2: If x = - 4
x² = (- 4) × (- 4) = 16

Example 3: If x = 0
x² = 0 × 0 = 0

Example 4. If x = 1.5
x² = 1.5 × 1.5 = 2.25

Example 5. If x = 2/5
x² = (2/5)×(2/5)=4/5

Perfect squares are products of an integer multiplied by itself. So, if x is an integer, then x2 is a perfect square.
Given below is a chart of the first 50 perfect squares for your reference.
 

The square root of a variable x is the number that, upon multiplying by itself, gives x as the product. It is written as √x. The square root of x²  is |x| because x × x=(-x) × (-x)=x2
So, √x² = x or x
For example, if x = - 7
√x² =√(-7)2 =√49 =7=7 or 7

x squared and 2x do not mean the same thing. In x², x is the base and 2 is the exponent. In 2x, 2 is the coefficient of x, indicating that x is multiplied by 2. 
For example, If x = 11, 
then x2 = x × x = 11 × 11 = 121, and 
2x = 2 × x = 2 × 11 = 22

The sum of squares can be found using the formula,
 a2+b2=(a+b)2-2ab
For instance, let’s take a = 3 and b = 4,
Substituting the values in the formula provided above, we get,
32 + 42 = (3 + 4)2 - 2(3)(4)
= 49 - 24
= 25

To find the difference of squares, we use the formula,
a2-b2=(a+b)(a-b) 
Let’s assume that a is 6 and b is 2
Substituting the values, we get,
62 - 22 = (6 + 2)(6 - 2) 
= 8 × 4
= 32
 

The formula for the difference of two perfect squares helps simplify complex algebraic expressions. It is useful while writing algebraic expressions as factors.
Let’s say there are two numbers x and y. To calculate the difference of their squares, we use the formula,
x2 - y2 = (x + y)(x - y)
Let us take x = 2 and y = 7
22 - 72 = (2 + 7)(2 - 7)
= 9 × (-5)
= -45

Completing the square refers to expressing a quadratic expression ax² + bx + c as a(x + d)² + e.
Where
a is the coefficient of x2 from the original expression.
d comes from b/2a, so (x + d)² is the part that is being completed.
e is the adjustment constant added to balance the expression after completing the square. It is calculated as e = c - a(b/2a)² =c-b²/4a.
For example, let’s complete the square for 2x2 + 8x + 5
Step 1: Factor out the coefficient of x²
= 2(x² + 4x) + 5
Step 2: Complete squaring inside parentheses 
42= 2, 2² = 4
Now add and subtract 4 inside the parentheses
= 2(x2 + 4x + 4 - 4) + 5 
= 2((x + 2)² - 4) + 5 
= 2(x + 2)² - 8 + 5 
= 2(x + 2)² - 3

The completed square form of 2x² + 8x + 5 is 2(x + 2)² - 3
Here, a = 2, d = 2 and e = - 3.
 

X squared as a mathematical concept has many real-life applications. From architecture to astronomy, x squared is used in many fields, some of which are discussed here:

• Area of a square: For a square with a side x, its area is x2. This is helpful in area related problems in geometry and architecture.

• Formula for kinetic energy : In physics, kinetic energy is calculated as KE = 12mv2, where v2 is the square of the object’s speed. This helps in understanding motion, designing vehicles, and improving crash safety.  

• Projectile motion: When we throw a football, the distance traveled and height achieved can be modeled using squared terms of time and velocity. This helps athletes, engineers, and game designers predict how things move through the air.

• Stress and strain calculations: In engineering/material science, stress and strain involve squared terms to model deformation under weight accurately.

• Satellite and signal coverage: To find the area a satellite or signal can cover, we use the formula for the area of a circle: A = r2. This is useful in planning GPS coverage, Wi-Fi range, and telecom networks.