Equations

An Equation is a combination of one or more terms separated with equal to “=” symbol. Terms can be numerical, alphanumerical, expression etc.  There are various methods of solving an equation. These include the Trial and Error Method, The Taylor method, Numerical Method, Solving Equations using Inverse Functions or principals of Elementary Algebra. Problems involving equations are required across grades and levels. Quadratic equations, Linear equations, Polynomial equations and Differential equations need a thorough understanding of basic and advanced Math concepts by students.

Solving an equation: A number which satisfies the given equation is called a solution (or root) of that.‘Satisfying the equation’ means that if the variable (literal, for example “x”…don’t look the nice picture on your left 🙂 involved in this is replaced by the number, then both sides of that become equal (identity).The process of finding the particular value of the variable which makes both sides of the equation equal is called solving the equation. Below are rules:

Rule 1:If equals are added to equals, the sums are equals. You can add the same number to both sides of an equation.

Rule 2: If equals are subtracted from equals, the remainders are equal. You can subtract the same number from both sides of an equation.

Rule 3:The two sides of an equation may be multiplied by same non-zero number.

For example, if x/2= 14, then x/2× 2 = 14× 2

Rule 4:The two sides of an equation may be divided by same non-zero number.

Below stated are the different types of equations.

Linear equations: Linear equation is any equation that when graphed produces a straight line.

To solve linear equations, you add, subtract, multiply and divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other side.

As long as you always do the same thing to BOTH sides of the equation, and do the operations in the correct order, you will get to the solution.

– Differential equations: An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a Differential Equations.

Partial differential equations: A partial differential equation is an equation which contains one or more partial derivatives. The order of the partial differential equation is that of the derivative of highest order in the equations.

In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations. They may sometimes be solved using a Bäcklund transformation, characteristics, Green’s function, integral transform, Lax pair, separation of variables, or numerical methods such as finite differences.

Quadratic equations: The equations that are having the highest degree term x2 or second degree are called as Quadratic Equations. It is an equation that looks like this:

$ax^2+bx+c=0$

where a, b, and c are numbers, called coefficients.

Example: x2+3x+4 = 0

You can think about a quadratic equation in terms of a graph of a quadratic function, which is called a parabola.

To solve a quadratic equation, you have to calculate a number called discriminant, usually denoted as d:

d = b2-4ac

Depending on the value of d, there are the folowing three possibilities:

– Discriminant d is greater than zero. The equation has two solutions.

– Discriminant is zero. There is only one solution.

$x = \frac{-b}{2a}$

– Discriminant is less than zero. No solutions are defined.

Note: for those of you who study complex numbers, there is a complex solution. If you do not know what complex numbers are, skip this part.

Radical equations: An equation in which the unknown appears in a radicand is called a radical equation. To solve a radical equation, begin by isolating the most complicated radical expression on one side of the equation, and then eliminate the radical by raising both sides of the equation to a power equal to the index of the radical. You may have to repeat this technique in order to eliminate all radical. When the equation is radical-free, simplify and solve it. Since extraneous roots may be introduced when both sides of an equation are raised to even powers, all roots must be checked by substitution in the original equation whenever a radical with an even index is involved.

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