Algebraic Equations

Algebra is a vast subject that studies about the algebraic expressions and equations. An algebraic expression is the combination of constants and variables; where, constants are the fixed quantities and variables are quantities that susceptible to vary. Equations are found everywhere in mathematics. An algebraic equation refers to an algebraic expression with a symbol of equality (=). In other words, an equation is an expression which has an equal to (=) sign between the two algebraic quantities or a set of quantity.
For example - Few algebraic equations are
(1) 4m2${}^2$n2${}^2$ + 2m = 0
(2) 7a + 4b + 9c = - 8
(3) 2 sin A + cos A = 2cos2${}^2$ A

Students may come across several different types of algebraic equations :
(1) Linear equation
(2) Quadratic equation
(3) Polynomial equation
(4) Trigonometric equation
(5) Radical equation
(6) Exponential equation

Solving equations means to find the value or set of values of unknown variable contained in it. Let us go ahead in this chapter and learn more about different types of differential equations and sample problems based on those.

Different Types of Equations

An equation is an important part of calculus, which comes under mathematics. It helps us to solve many problem. 

Types of Algebraic Equations

Lets discuss different types of algebraic equations:

1) Linear Equations:

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The graph of linear equation is a straight line if there are two variables.

General form of the linear equation with two variables:

y = mx + c, m ≠$\ne$ 0.

Where, m is known as slope and c at that point on which it cut y axis.

Linear equations with different variables:

a) The equation with one variable:

An equation who have only one variable.

Examples: 

1. 8a - 8 = 0
2. 9a = 72.

b) The equation with two variables: 

An equation who have only two types of variable in the equation.

Examples:

1. 7x + 7y = 12
2. 8a - 8d = 74
3. 9a + 6b - 82 = 0.

c) The equation with three variables:

An equation who have only three types of variable in the equation.

Examples: 

1. 5x + 7y - 6z = 12
2. 13a - 8b + 31c = 74
3. 6p + 14q -7r + 82 = 0.

2) Radical Equations:

An equation whose maximum exponent on the variable is 12$\frac{1}{2}$ and have more than one term or we can say that a radical equation is an equation in which the variable is lying inside a radical symbol usually in a square root.

Examples:

1. x−−√+10=26$\sqrt{x}+10=26$
2. x2−5−−−−−√+x−1$\sqrt{x^2-5}+x-1$

3) Quadratic Equations:

Quadratic equation is the second degree equation in one variable contains the variable with an exponent of 2.

The general form:

ax2 + bx + c = 0, a ≠$\ne$ 0

Examples of Quadratic Equations:

1. 8x2 + 7x - 75 = 0
2. 4y2 + 14y - 8 =0
3. 5a2 - 5a = 35

4) Exponential Equations:

An equation who have variables in the place of exponents. Exponential equation can be solved using the property: ax$a^x$ = ay$a^y$ => x = y.

Examples: 

1. ab = 0 Here "a" is base and "b" is exponent.
2. 4x =0
3. 8x = 32.

5) Rational Equations:

A rational equation is one that involves rational expressions.

Example:

x2$\frac{x}{2}$ = x+24$\frac{x+2}{4}$.

Types of Math Equations

Here, we will solve some algebraic equations:

Solved Examples

Question 1: Solve the quadratic equation x2 + 6x - 27 = 0.
Solution:

Given quadratic equation: x2 + 6x - 27 = 0

x2 + 6x - 27 = 0

x2 + 9x - 3x - 27 = 0

=> x(x + 9) - 3(x - 9) = 0

=> (x - 3)(x + 9) = 0

=> either x - 3 = 0 or x + 9 = 0

=> x = 3 or x = -9

The values of x are (3, -9).

Question 2: Solve 8x = 32
Solution:

Given equation is 8x = 32.

=> 8x = 32

or (23)x = 25

or 23x = 25

Since the bases of the equation are same, set the exponents equal to one another:

=> 3x = 5

x = 53$\frac{5}{3}$.

Question 3: Solve x+3−−−−√$\sqrt{x+3}$ = 2
Solution:

Given equation is x+3−−−−√$\sqrt{x+3}$ = 2

=> x+3−−−−√$\sqrt{x+3}$ = 2

To isolate the x, squaring both side

=> x + 3 = 4

x = 4 - 3

x = 1.