Radical Expression

Definition

 A radical expression is an expression containing a square root.

Examples of Radical Expression

are examples of radical expression.

It is also an example of  Redical Expression

To solve this kind of equations, remember when you have a variable inside a square root, you undo the root by doing the opposite: squaring.

Example: Given , you would square both sides:

More about Radical Expression

Radical: Thesymbol that is used to denote square root or nth roots.
Radicand: Radicand is a number or expression inside the radical symbol.

For example, 5 is the radicand in.

Radical equation: An equation containing radical expressions with variables in the radicands.
Radical inequality: An inequality containing a radical expression with the variable in the radicand.

Solved Example on Radical Expression

Evaluate the radical expression  when a = 2 and b = 4.
Choices:
A. 9
B. 8
C. 7
D. 6
Correct Answer: D
Solution:
Step 1:         [Substitute the values of a and b in the given radical expression.]
Step 2:                                    [Find the positive square root.]
Step 3:                                        [Multiply.]
Step 4:                                           [Add.]
Step 5: = 6                                            [Simplify.]

The terms which are Related for Redical Expression

• Expression
• Square Root
• Variable
• Radicand
• Radical
• Radical Equation
• Radical Inequality 

 Note: You can always check your answers in a solved equation by plugging your answer back into the original equation and making sure that it fits. (This is because the very act of squaring the sides can create solutions that never existed before).

Radical Expression

By Zaib Abbasi 23:39

Completing the Square

"Completing the Square" is where you ... 

... take a Quadratic Equation like this:		and turn it into this:
ax2 + bx + c = 0		a(x+d)2 + e = 0

For those of you in a hurry, I can tell you that:
and:

But if you have time, let me show you how to "Complete the Square" yourself.

Completing the Square

Say we have a simple expression like x² + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x² + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)²

In Algebra it looks like this:

x2 + bx		+ (b/2)2	=	(x+b/2)2
		"Complete the Square"

So, by adding (b/2)² we can complete the square.

And (x+b/2)² has x only once, which is easier to use.

Keeping the Balance

Now ... you can't just add (b/2)² without also subtracting it too! Otherwise the whole value would change.

So I will show you how to do it properly with an example:

Start with:
	("b" is 6 in this case)
Complete the Square:	Also subtract the new term
Simplify it and we are done.

The result:

x² + 6x + 7   =   (x+3)² - 2

And now x only appears once, and your job is done!

Now, let us look at a useful application: solving Quadratic Equations ...

Solving General Quadratic Equations by Completing the Square

We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have a coefficient of a in front of x²:

ax² + bx + c = 0

But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:

x² + (b/a)x + c/a = 0

Steps

Now we can solve a Quadratic Equation in 5 steps:

• Step 1 Divide all terms by a (the coefficient of x²).
• Step 2 Move the number term (c/a) to the right side of the equation.
• Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p)² = q, which can be solved rather easily:

• Step 4 Take the square root on both sides of the equation.

• Step 5 Subtract the number that remains on the left side of the equation to find x.

Examples

I have two examples to show you how:

Example 1: Solve x² + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x² is 1

Step 2 Move the number term to the right side of the equation:

x² + 4x = -1

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.

(b/2)² = (4/2)² = 2² = 4

x² + 4x + 4 = -1 + 4

(x + 2)² = 3

Step 4 Take the square root on both sides of the equation:

x + 2 = ±√3 = ±1.73 (to 2 decimals)

Step 5 Subtract 2 from both sides:

x = ±1.73 – 2 = -3.73 or -0.27

And here is an interesting and useful thing. At the end of step 3 we had the equation: (x + 2)2 = 3 It gives you the vertex (turning point) of x2 + 4x + 1: (-2, -3)

Example 2: Solve 5x² – 4x – 2 = 0

Step 1 Divide all terms by 5

x² – 0.8x – 0.4 = 0

Step 2 Move the number term to the right side of the equation:

x² – 0.8x = 0.4

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2)² = (0.8/2)² = 0.4² = 0.16

x² – 0.8x + 0.16 = 0.4 + 0.16

(x – 0.4)² = 0.56

Step 4 Take the square root on both sides of the equation:

x – 0.4 = ±√0.56 = ±0.748 (to 3 decimals)

Step 5 Subtract (-0.4) from both sides (in other words, add 0.4):

x = ±0.748 + 0.4 = -0.348 or 1.148

Why "Complete the Square"?

Why would you want to complete the square when you can just use the Quadratic Formula to solve a Quadratic Equation?

Well, one reason is given above, where the new form not only shows you the vertex, but makes it easier to solve.

There are also times when the form ax² + bx + c may be part of a larger problem and rearranging it asa(x+d)² + e makes the solution easier, because x only appears once.

For example "x" may itself be a function (like cos(z)) and rearranging it may open up a path to a better solution.

Also Completing the Square is the first step in the Derivation of the Quadratic Formula

Just think of it as another tool in your mathematics toolbox.

Completing the Square

By Zaib Abbasi 03:08

Derivation of Quadratic Formula

A Quadratic Equation looks like this:
                                      
A Qaudratic Formula lookd like as
                                        
The Quadratic Formula looks like a magic,but we can can follow these steps to see how it comes

1. Complete the Square

It is hard to handle an equation where "x" appears twice, but there is a way to rearrange it so that "x" only appears once. It is called "Completing the Square" (please read that first!).

So, let's go:

Start with
Divide the equation by a
Put c/a on other side
Add (b/2a)2 to both sides
The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a" So we can re-write it this way:
"Complete the Square"

2. Now Solve For "x"

Now we just need to rearrange the equation to leave "x" on the left

Start with
Square root
Move b/2a to right
That is actually solved! But let's simplify it a bit:
Multiply right by 2a/2a
Simplify:
Which is the Quadratic formula we all know and love:

Derivation of Quadratic Formula

By Zaib Abbasi 23:02

Quadratic Equations

The name of  the this equations i.e Quadratic comes from "Quads" which means square because variable gets squared (like x²).
It is also known as Equation of Degree 2" (because of the "2" on the x)

Here is the Standard Form of a Quadratic Equation looks like this:

                                                

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (you don't know it yet)

An example of a Quadratic Equation:

                                        
Here we are giving some more Examples

          In this one a=2, b=5 and c=3

                   This one is a little more tricky:
                                        

• Where is a? In fact a=1, as we don't usually write "1x²"
• b = -3
• And where is c? Well, c=0, so is not shown.

     Oops! This one is not a quadratic equation, because it is missing x² (in other words a=0, and that means it can't be quadratic)

Quadratic Equations

By Zaib Abbasi 23:01