In this mini-lesson, we will explore what is x squared, the difference of squares, and solving quadratic by completing the squares.

In algebra, we commonly come across the term x squared. Do you aware of what is x squared?

We are going to learn particularly about (x^2) in this mini-lesson.

**Lesson Plan**

x squared is a notation that is used to represent the expression (xtimes x).

i.e., x squared equals x multiplied by itself.

In algebra, (x) multiplied by (x) can be written as (xtimes x) (or) (xcdot x) (or) (x, x) (or) (x(x))

(x) squared symbol is (x^2).

Here:

- (x) is called the base.
- 2 is called the exponent.

(x) squared = (x^2) = (xtimes x)

Here are some examples to understand (x) squared better.

Phrase Expression

x squared times x

(x^2times x =x^3)

x squared minus x

(x^2-x)

x squared divided by x

(x^2div x =x^1=x)

x squared times x squared

(x^2times x^2 =x^4)

x squared plus x squared

(x^2+x^2 =2x^2)

x squared plus y squared

(x^2+y^2) square root x2

(sqrt{x^2}=x)

x squared times x cubed

(x^2times x^3 =x^5)

Here is the completing the square calculator. We can enter any quadratic expression here and see how the square can be completed..

**Example 1**

Can we help Sophia to understand (x^2) and (2x) don’t need to be the same by evaluating them at (x= -6)?

**Solution**

It is given that (x=-6).

Then:

[begin{align} x^2 &= (-6)^2 = -6 times -6 = 36\[0.2cm] 2x &= 2(-6) = 2 times -6 = -12 end{align}]Here, (x^2 neq 2x).

Therefore,

(x^2) and (2x) don’t need to be the same **Example 2**

Can we help Jim to factorize the following expression using the formula of difference of squares?

[x^4-16]**Solution**

The formula of difference of squares says: [x^2-y^2=(x+y)(x-y)]

We will apply this to factorize the given expressions as many times as needed.

[begin{align} x^4-16 &= (x^2)^2 – 4^2\[0.2cm] &= (x^2+4)(x^2-4)\[0.2cm] &=(x^2+4)(x^2-2^2)\[0.2cm] &=(x^2+4)(x+2)(x-2) end{align}]Therefore, the given expression can be factorized as

((x^2+4)(x+2)(x-2)) **Example 3**

The area of a square-shaped window is 36 square inches. Can you find the length of the window?

**Solution**

Let us assume that the length of the window is (x) inches.

Then its area using the formula of area of a square is ( x^2) square inches.

By the given information, [x^2 = 36]

By taking the square root on both sides, [ sqrt{x^2}= sqrt{36}]

We know that the square root of (x^2) is (x).

The square root of 36 is 6 because (6^2=36).

Therefore,

(therefore) The length of the window = 6 inches **Example 4**

Solve by completing the square.

[x^2-10x+16=0]**Solution**

The given quadratic equation is:

[x^2-10x+16=0]We will solve by completing the square.

Here, the coefficient of (x^2) is already (1)

The coefficient of (x) is (-10)

The square of half of it is ((-5)^2 =25)

Adding and subtracting it on the left-hand side of the given equation after the (x) term:

[ begin{aligned} x^2-10x+25-25+16&=0\[0.2cm](x-5)^2-25+16&=0\ [because x^2!-!10x!+!25!=! (x!-!5)^2 ]\[0.2cm] (x-5)^2-9&=0\[0.2cm] (x-5)^2& =9 \[0.2cm] (x-5) &= pmsqrt{9} \ [ text{Taking square root }&text{on both sides} ]\[0.2cm] x-5=3; ,,,,&x-5= -3\[0.2cm] x=8; ,,,,&x = 2 end{aligned} ](therefore) (x=8,, , 2)