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# Mathematical Expressions

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 Sub Topics Mathematical expression is a series of numbers which are joined together with different mathematical operators. In order to solve the series of these numbers, we need to know a particular order which is to be followed to get the solution to the mathematical expressions. Else, every time the mathematical expression is solved, you will get a different result.The rule PEMDAS is to be remembered, when we are solving the set of mathematical operators. Here, ‘P’ stands for Parenthesis. So, we will first do the operation on the Parenthesis. Once the Parenthesis is solved, we say that ‘E’ is to be looked for, which means 'Exponents'. After this, we have ‘M’, which means Multiplication. So, the operator of multiplication is to be performed, if it exists in the expression. After performing the multiplication operator, we will look for the ‘D’, which stands for the Division operator. So, we have to perform the operation of division after the multiplication operator. After the operation of division and multiplication, we have ‘A’ which stands for Addition, which is followed by ‘S’ which stands for Subtraction. We will perform the operation of addition in the mathematical expression. Once the addition operator is performed, we are left with the subtraction operation. In this way, we get the accurate result of the mathematical expressions, which is global every time.

## What is a Mathematical Expression?

Mathematical expression is defined as a mixture of arithmetic operations, operated on variables and constants. We can have more than one arithmetic operator in the expression. Usually, operations are like addition, subtraction, multiplication and division.

Expression contain variables and are generally represented by english alphabets like p, q, x, y, a, b etc. The expressions can be of several types and have different descriptions for different representations of the relationships of the variables and the constants. Variable represents any unknown quantity that holds certain numeric value and has to be evaluated in most of the mathematical expressions.

In different expressions, we may use the same name of the variable. But, the values can differ from one expression to another.

## Mathematical Expressions and Equations

Mathematical expression represents a numeric value and an expression does not contain an 'equal to' (=) sign. It is a string of mathematical symbols placed on one side of an equation and an expression will have no solution. Two most common types of mathematical equation is arithmetic and algebraic equation.

Examples of mathematical expressions are 5 + 5, (88 - 15) + 8, $\sqrt{569}$ + 2, 3xy + 7y etc.,

Mathematical equation is a mathematical sentence that says two things are equal. It can be solved and indicates the equality of two expressions, sequence of symbols is split into left and right sides joined by an equal sign.

Examples of mathematical equations are 3x + 5y = 12, x + y + z = 10, $\frac{7}{5}$x + $\frac{1}{3}$ z = 5

## Examples of Mathematical Expressions

Given below are some of the examples of mathematical expressions.

### Solved Examples

Question 1: Solve 3x$^{2}$ + 2x + 4 = 0.
Solution:
Step - 1:
Coefficients of a, b and c are 3, 2 and 4 respectively.

x = $\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Step - 2:
Plug in the values of a, b and c in the above formula.

x = $\frac{-2\pm\sqrt{2^{2}-4 \times 3 \times 4}}{2 \times 3}$

Step - 3:
Simplify the expression under the square root.

x = $\frac{-2\pm\sqrt{-44}}{6}$

Solving for x, we get

$x_{1}$ = $\frac{-2+\sqrt{-44}}{6}$

$x_{1}$ = -$\frac{1}{3}$ + $\frac{1}{3}$ $\sqrt{11i}$

$x_{2}$ = $\frac{-2-\sqrt{-44}}{6}$

$x_{2}$ = -$\frac{1}{3}$ - $\frac{1}{3}$ $\sqrt{11i}$

Step - 4:
Therefore, the solutions for the given equation are as follows:

$x_{1}$ = -$\frac{1}{3}$ + $\frac{1}{3}$ $\sqrt{11i}$ and $x_{2}$ = -$\frac{1}{3}$ - $\frac{1}{3}$ $\sqrt{11i}$

Question 2: Evaluate 2x$^{4}$ + 3x$^{3}$ - 5x$^{2}$ + 7y for x = -2, y = 1.
Solution:
Given: 2x$^{4}$ + 3x$^{3}$ - 5x$^{2}$ + 7y
In the given expression, plug in x = -2 and y = 1
= 2(- 2)$^{4}$ + 3(- 2)$^{3}$ - 5(- 2)$^{2}$ + 7(1)
= 2 x 16 - 24 - 20 + 7
= 39 - 44
= -5