Variable and Expression

In the previous post we have discussed about Completing the Square and In today's session we are going to discuss about Variable and Expression. When we study about the algebra and finding the solution of the algebraic equations, we say that the child must have the knowledge of the variable and expression before we start the study of the equations. By the term variable, we mean the values which are unknown and the different values can be placed to these variables. Now we will say that the expression can be formed by joining the terms with the operators + and -.
So we say that x , y , z are the variables. On the other hand we have 3x, 5y, 6z as the terms which are used as the units for the formation of the algebraic expressions. So we say that 3x + 5y – 2z is the expression. We can perform the mathematical operations on the expressions.

In order to add the expressions, we need to remember that the like terms of the expressions can be added together. So if we have the two expressions 3x + 6y and 4x – 2y so to add the given expressions we will proceed as follows:
(3x + 6y) + (4x – 2y),
= ( 3x + 4x ) + ( 6y – 2y),
= 7x + 4y.
Thus we say that the coefficient of the like terms are added up. In the same way the operation of subtraction can also be performed. So if we take the two expression: 3x + 6y and 4x – 2y . If we need to find the difference of the two expressions we will write it as follows: (3x + 6y) – ( 4x – 2y).
= ( 3x – 4x ) + ( 6y + 2y),
= -x + 8y.
We can also perform the operation of multiplication and division on the algebraic expressions. Different rules are followed for solving such expressions.
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Completing the Square

In this blog we will see how to Completing the Square. To find the complete square of any term first we need to take a quadratic equation. For example: as we know that the equation which has highest power is 2 is said to be quadratic equation. The general form can be written as: ax² + bx + c = 0, then change the general form into this given form:

a ( x + d)² + e = 0.

If we want to find the value of variable ‘d’ then we use formula: the formula to find the vaule of ‘d’ given as: d = b / 2a and formula to find the vlaue of ‘e’ is given as: e = c – b² / 4a. To calculate the complete square we need to follow some basic steps so that we can easily understand the process.

Step 1: First of all we have to take a simple expression. Suppose we have simple expression as: x² + bx. Here in this expression value of x is twice.

Step 2: Then complete square in the given expression. So it can be written as: x² + bx + (b / 2)² = (x + b / 2)², it means if we add (b / 2)² then we get the square.

Let understand all the steps with the help of small example:

For example: p² + 6x + 7, in this expression the value of b is 6,

Solution: Given expression is p² + 6x + 7, to find its complete square we need to follow the above mention steps:

First of all we have to find the complete square or value of d. we can find the vlaue of d using the above given formula:

d = b / 2a = 6 / 2

So it can be written as:

p² + 6p + (6 / 2)² + 7 – (6 / 2)²

So it can be written as:

(p + 6 / 2)² + 7 – 9 on further solving we get:

(p + 3)² – 2. in this way we can find out the complete square.

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