X Squared

What is x squared?

 

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.

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:

Here are some examples to understand it better.

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.

Solved Examples

 

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.

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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