X Squared
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
What is x squared?
x squared is a notation that is used to represent the expression \(x\times x\).
i.e., x squared equals x multiplied by itself.
In algebra, \(x\) multiplied by \(x\) can be written as \(x\times x\) (or) \(x\cdot x\) (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 = \(x^2\) = \(x\times x\) |
Here are some examples to understand \(x\) squared better.
| Phrase | Expression |
|---|---|
|
x squared times x |
\(x^2\times x =x^3\) |
|
x squared minus x |
\(x^2-x\) |
|
x squared divided by x |
\(x^2\div x =x^1=x\) |
|
x squared times x squared |
\(x^2\times x^2 =x^4\) |
|
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 |
\(x^2\times x^3 =x^5\) |
- Here we use the laws of exponents in case of multiplying or dividing the exponents of the same base.
\[\begin{aligned}
x^{m} \cdot x^{n} &=x^{m+n} \\
\frac{x^{m}}{x^{n}} &=x^{m-n}
\end{aligned}\] - The formulas for the squares of the sum and the difference are:
\[\begin{array}{l}
(x+y)^{2}=x^{2}+2 x y+y^{2} \\
(x-y)^{2}=x^{2}-2 x y+y^{2}
\end{array}\]
Is x Squared Same as 2x?
No, \(x^2\) is NOT same as \(x\).
Using the exponents, \(x^2 = x \times x \).
But \(2x = 2 \times x= x + x\), because multiplication is nothing but the repeated addition.
Here are some examples to understand it better.
|
\(x\) |
\(x^2 = x \times x\) | \(2x = 2 \times x\) |
|---|---|---|
| 3 |
\(3 \times 3=9\) |
2(3) = 6 |
| -1 |
\(-1 \times -1 = 1\) |
2(-1) = -2 |
| -2 |
\(-2 \times -2 =4\) |
2(-2) = -4 |
Special Factoring: Difference of Squares
While factoring algebraic expressions, we may come across an expression that is a difference of squares.
i.e., an expression of the form \(x^2-y^2\).
There is a special formula to factorize this:
| \(x^2-y^2=(x+y)(x-y)\) |
Here are some examples to understand it better.
| \(x^2-y^2\) |
\((x+y)(x-y)\) |
|---|---|
|
\(x^2-3^2\) |
\((x+3)(x-3)\) |
|
\(y^2-x^2\) |
\((y+x)(y-x)\) |
|
\(x^2-4y^2\) |
\((x+2y)(x-2y)\) |
Solving Quadratics by Completing the Square
Completing the square in a quadratic expression \(ax^2+bx+c\) means expressing it of the form \(a(x+d)^2+e\).
Let us learn how to complete a square using an example.
Example
Complete the square in the expression
\[-4 x^{2}-8 x-12\]
Solution:
First, we should make sure that the coefficient of \(x^2\) is \(1\)
If the coefficient of \(x^2\) is NOT \(1\), we will place the number outside as a common factor.
We will get:
\[-4 x^{2}-8 x-12 = -4 (x^2+2x+3)\]
Now, the coefficient of \(x^2\) is \(1\)
Step 1: Find half of the coefficient of \(x\)
Here, the coefficient of \(x\) is \(2\)
Half of \(2\) is \(1\)
Step 2: Find the square of the above number
\[1^2=1\]
Step 3: Add and subtract the above number after the \(x\) term in the expression whose coefficient of \(x^2\) is \(1\)
\[\begin{align} -4 (x^2\!+\!2x\!+\!3)\!&=\!\!-4 \left(x^2\!+\!2x\! +\color{green}{\mathbf{1 -1}} \!+\!3 \right)\end{align}\]
Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the identity \( x^2+2xy+y^2=(x+y)^2\)
In this case, \[x^2+2x+ 1= (x+1)^2\]
The above expression from Step 3 becomes:
\(-4 \left(\color{green}{x^2\!+\!2x \!+\!1\!}-\!1 \!+3\right)\)=\(-4 (\!\color{green}{(x+1)^2}\!\! -\!1+3\!)\)
Step 5: Simplify the last two numbers.
Here, \(-1+3=2\)
Thus, the above expression is:
\[ -4 (x+1)^2 \color{green}{-1+3} = -4 ((x+1)^2 +\color{green}{2}) \\= -4(x+1)^2-8\]
This is of the form \(a(x+d)^2+e\).
Hence, we have completed the square.
| Thus, \(-4 x^2-8 x-12= -4 (x+1)^2 -8)\) |
Here is the completing the square calculator. We can enter any quadratic expression here and see how the square can be completed..
Solved Examples
| 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:
\[ \begin{aligned} x^2-10x+25-25+16&=0\\[0.2cm](x-5)^2-25+16&=0\\ [\because x^2\!-\!10x\!+\!25\!=\! (x\!-\!5)^2 ]\\[0.2cm] (x-5)^2-9&=0\\[0.2cm] (x-5)^2& =9 \\[0.2cm] (x-5) &= \pm\sqrt{9} \\ [ \text{Taking square root }&\text{on both sides} ]\\[0.2cm] x-5=3; \,\,\,\,&x-5= -3\\[0.2cm] x=8; \,\,\,\,&x = 2 \end{aligned} \]
| \(\therefore\) \(x=8,\, \, 2\) |
- Solve by completing the square.
\[x^4-18 x^2+17=0\]
Hint: Assume \(x^2=t\) - Write the following equation of the form \((x-h)^2+(y-k)^2=r^2\) by completing the square.
\[x^2+y^2-4 x-6 y+8=0\]
Hint: Group \(x\) terms separately and \(y\) terms separately and then complete the squares.
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The mini-lesson targeted the fascinating concept of x squared. We explored x squared, x squared equals, square root, x cubed, what is x Squared x, x 2, x squared times x, x squared plus x squared, x squared symbol, x squared minus x, x squared divided by x, and x squared plus y squared.
The math journey around x squared starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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Frequently Asked Questions (FAQs)
1. What is a squared minus b squared?
This is given by the difference of squares formula:
\[a^2-b^2=(a+b)(a-b)\]
2. What does 3 x squared mean?
3 x squared means \(3x^2\).
Its 3 times \(x^2\).
3. How do you type 2 x squared?
2 x squared can be typed as \(2x^2\).
Here, the 2 above \(x\) is a superscript.
4. How do you find square root?
To find the square root of a number, we have to see by multiplying which number by itself, we can get the given number.
For example,
\[ \sqrt{9} = \sqrt{3^2} = 3\]