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)