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 x^{2} |
\(\sqrt{x^2}=x\) |

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.

**About Cuemath**

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

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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.

**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,

\[ \sqrt{9} = \sqrt{3^2} = 3\]