# X Squared

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

 1 What is x Squared? 2 Important Notes on x Squared 3 Solved Examples on x Squared 4 Challenging Questions on x Squared 5 Interactive Questions on x Squared

## 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

$$x^2\times x^3 =x^5$$

Important Notes
1. 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}
2. 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}

 $$\therefore$$ $$x=8,\, \, 2$$

Challenging Questions
1. Solve by completing the square.
$x^4-18 x^2+17=0$
Hint: Assume $$x^2=t$$
2. 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.

## About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 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,

$\sqrt{9} = \sqrt{3^2} = 3$

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