What Does X Squared Equal

In this mini-lesson, we will explore what is x squared, the divergence of squares, and solving quadratic by completing the squares.

In algebra, we commonly come across the term x squared. Practise you enlightened of what is x squared?

What is x squared?

Nosotros are going to learn particularly about \(x^two\) in this mini-lesson.

Lesson Programme

What is 10 squared?

x squared is a notation that is used to stand for the expression \(x\times x\).

i.e., x squared equals 10 multiplied past itself.

In algebra, \(x\) multiplied by \(x\) can be written as \(10\times ten\) (or) \(x\cdot x\) (or) \(x\, ten\) (or) \(x(x)\)

\(10\) squared symbol is \(ten^2\).

Here:

\(x\) is called the base.
2 is called the exponent.

\(x\) squared = \(10^2\) = \(ten\times x\)

Hither are some examples to understand \(x\) squared better.

Phrase	Expression
x squared times x	\(ten^two\times x =x^three\)
x squared minus x	\(ten^ii-x\)
x squared divided by x	\(ten^2\div x =x^i=x\)
ten squared times x squared	\(ten^2\times x^ii =x^4\)
10 squared plus x squared	\(x^two+10^2 =2x^2\)
10 squared plus y squared	\(x^2+y^ii\)
square root x²	\(\sqrt{ten^two}=x\)
x squared times 10 cubed	\(10^2\times x^3 =ten^v\)

important notes to remember

Important Notes

Here nosotros utilize the laws of exponents in case of multiplying or dividing the exponents of the same base.
\[\begin{aligned}
x^{m} \cdot 10^{north} &=x^{m+n} \\
\frac{10^{m}}{x^{due north}} &=x^{m-n}
\end{aligned}\]
The formulas for the squares of the sum and the deviation are:
\[\brainstorm{array}{50}
(ten+y)^{2}=x^{ii}+two x y+y^{2} \\
(x-y)^{two}=x^{2}-two 10 y+y^{2}
\end{assortment}\]

Is 10 Squared Same as 2x?

No, \(ten^2\) is Not same as \(x\).

Using the exponents, \(10^2 = 10 \times x \).

But \(2x = 2 \times x= x + x\), considering multiplication is nothing but the repeated improver.

Is x squared same as 2x?

Here are some examples to sympathize it amend.

\(x\)	\(x^2 = x \times 10\)	\(2x = 2 \times x\)
3	\(3 \times iii=9\)	2(3) = 6
-1	\(-1 \times -1 = ane\)	2(-one) = -2
-2	\(-ii \times -2 =4\)	ii(-ii) = -4

Special Factoring: Difference of Squares

While factoring algebraic expressions, nosotros may come beyond an expression that is a difference of squares.

i.e., an expression of the form \(x^two-y^2\).

There is a special formula to factorize this:

Here are some examples to understand it amend.

\(x^2-y^2\)	\((x+y)(x-y)\)
\(x^2-three^2\)	\((x+three)(x-3)\)
\(y^2-ten^2\)	\((y+x)(y-x)\)
\(10^2-4y^ii\)	\((x+2y)(10-2y)\)

Solving Quadratics by Completing the Square

Completing the square in a quadratic expression \(ax^ii+bx+c\) means expressing it of the form \(a(x+d)^2+e\).

Introduction to how to complete the square is explained using a quadratic equation.

Let us learn how to complete a square using an example.

Example

Complete the foursquare in the expression

\[-4 x^{2}-8 10-12\]

Solution:

First, we should make sure that the coefficient of \(x^2\) is \(one\)

If the coefficient of \(x^ii\) is NOT \(1\), nosotros volition place the number outside every bit a common cistron.

Nosotros will get:

\[-4 x^{2}-8 ten-12 = -4 (x^2+2x+iii)\]

At present, the coefficient of \(x^two\) is \(i\)

Stride one: Find half of the coefficient of \(x\)

Here, the coefficient of \(x\) is \(two\)

Half of \(2\) is \(1\)

Stride 2: Observe the square of the in a higher place number

\[1^2=1\]

Step 3: Add and decrease the above number after the \(ten\) term in the expression whose coefficient of \(ten^ii\) is \(1\)

\[\begin{marshal} -4 (x^2\!+\!2x\!+\!3)\!&=\!\!-4 \left(x^ii\!+\!2x\! +\colour{green}{\mathbf{1 -i}} \!+\!three \right)\end{marshal}\]

Footstep 4: Factorize the perfect square trinomial formed by the outset three terms using the identity \( x^2+2xy+y^2=(x+y)^2\)

In this case, \[x^2+2x+ 1= (ten+ane)^2\]

The in a higher place expression from Step 3 becomes:

\(-4 \left(\color{green}{x^ii\!+\!2x \!+\!i\!}-\!1 \!+iii\right)\)=\(-4 (\!\color{green}{(x+ane)^2}\!\! -\!1+3\!)\)

Step five: Simplify the terminal two numbers.

Here, \(-one+3=two\)

Thus, the to a higher place expression is:

\[ -four (ten+i)^2 \color{green}{-1+3} = -4 ((x+one)^two +\color{light-green}{2}) \\= -4(x+1)^two-8\]

This is of the form \(a(x+d)^2+e\).

Hence, we have completed the square.

Thus, \(-4 x^two-8 x-12= -4 (x+i)^2 -eight)\)

Here is the completing the square reckoner. We can enter whatever quadratic expression here and run across how the square can be completed..

Solved Examples

Can we help Sophia to understand \(10^two\) and \(2x\) don't demand to be the same by evaluating them at \(x= -vi\)?

Solution

It is given that \(10=-half-dozen\).

Then:

\[\begin{align} x^2 &= (-six)^two = -6 \times -half-dozen = 36\\[0.2cm]
2x &= 2(-6) = 2 \times -6 = -12 \end{align}\]

Here, \(x^two \neq 2x\).

Therefore,

\(x^two\) and \(2x\) don't need to be the same

Tin can nosotros help Jim to factorize the following expression using the formula of divergence of squares?

\[x^iv-sixteen\]

Solution

The formula of difference of squares says: \[x^ii-y^2=(10+y)(x-y)\]

We will apply this to factorize the given expressions as many times as needed.

\[\begin{align}
x^4-xvi &= (10^2)^two - iv^2\\[0.2cm]
&= (x^2+4)(x^2-iv)\\[0.2cm]
&=(x^two+four)(x^two-2^2)\\[0.2cm]
&=(x^2+4)(10+ii)(x-2)
\terminate{align}\]

Therefore, the given expression can exist factorized equally

The area of a foursquare-shaped window is 36 square inches. Tin can y'all notice the length of the window?

x squared word problem: a square shaped window

Solution

Allow u.s. presume that the length of the window is \(10\) inches.

Then its expanse using the formula of area of a square is \( x^ii\) square inches.

Past the given data, \[10^2 = 36\]

By taking the square root on both sides, \[ \sqrt{x^2}= \sqrt{36}\]

Nosotros know that the square root of \(x^2\) is \(x\).

The square root of 36 is half-dozen because \(6^2=36\).

Therefore,

\(\therefore\) The length of the window = 6 inches

Solve by completing the square.

\[x^2-10x+16=0\]

Solution

The given quadratic equation is:

\[x^2-10x+sixteen=0\]

We volition solve past completing the square.

Here, the coefficient of \(ten^ii\) is already \(1\)

The coefficient of \(ten\) is \(-10\)

The foursquare of one-half of it is \((-5)^2 =25\)

Adding and subtracting it on the left-mitt side of the given equation after the \(x\) term:

\[ \begin{aligned} ten^2-10x+25-25+sixteen&=0\\[0.2cm](10-v)^2-25+16&=0\\ [\considering ten^2\!-\!10x\!+\!25\!=\! (10\!-\!five)^2 ]\\[0.2cm] (x-5)^2-9&=0\\[0.2cm] (10-5)^2& =nine \\[0.2cm] (10-v) &= \pm\sqrt{9} \\ [ \text{Taking square root }&\text{on both sides} ]\\[0.2cm] x-v=3; \,\,\,\,&x-5= -iii\\[0.2cm] x=8; \,\,\,\,&10 = ii \end{aligned} \]

\(\therefore\) \(10=8,\, \, two\)

Challenge your math skills

Challenging Questions

Solve by completing the square.
\[ten^4-18 x^two+17=0\]
Hint: Presume \(ten^2=t\)
Write the following equation of the form \((x-h)^2+(y-grand)^2=r^two\) by completing the foursquare.
\[ten^2+y^ii-4 ten-6 y+8=0\]
Hint: Grouping \(x\) terms separately and \(y\) terms separately and then complete the squares.

Interactive Questions

Here are a few activities for you to do.

Select/type your answer and click the "Check Reply" button to run into the issue.

Allow'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 ten, x 2, x squared times x, ten squared plus x squared, 10 squared symbol, ten squared minus x, x squared divided by x, and 10 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 fashion that not just information technology is relatable and like shooting fish in a barrel to grasp, just also will stay with them forever. Here lies the magic with Cuemath.

Well-nigh Cuemath

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

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

Be information technology worksheets, online classes, doubt sessions, or any other grade of relation, information technology'due south the logical thinking and smart learning arroyo that we, at Cuemath, believe in.

Oft Asked Questions (FAQs)

1. What is a squared minus b squared?

This is given by the difference of squares formula:

\[a^ii-b^2=(a+b)(a-b)\]

2. What does three 10 squared mean?

3 x squared ways \(3x^2\).

Its 3 times \(x^2\).

3. How do you type ii ten squared?

2 x squared can exist typed as \(2x^ii\).

Here, the ii to a higher place \(10\) is a superscript.

iv. How do you find square root?

To find the square root of a number, nosotros accept to see by multiplying which number by itself, nosotros can go the given number.

For case,

\[ \sqrt{ix} = \sqrt{iii^2} = 3\]