Standard form of a quadratic equation

• What is the Standard form of a quadratic equation?
• Importance of the standard form of a quadratic equation
• How to convert a quadratic equation to the standard form
• How to get the values of a, b and c from a quadratic equation in the standard form

Introduction

The quadratic equation can be written in different forms. One of these is known as the standard form of the quadratic equation.

This is a method of writing a quadratic equation so that all the terms in it are on one side and 0 is on the other side. Furthermore, in the standard form of the quadratic equation, the term containing an 'x^2' is written first, then the term containing an 'x' comes second, and the term without any variable (the constant) comes at last. If a, b and c are regarded as the numerical coefficients of these terms respectively, then all quadratic equations in the standard form have essentially the following structure:

ax² + bx + c = 0

Importance
The quadratic formula, the nature of the roots, the discriminant value are three quantities that are related to quadratic equations. The standard form of a quadratic equation is directly applicable for each of these as follows:

Value of Discriminant = sqrt(b² - 4ac)

Quadratic formula: x = (-b +/- sqrt(b² - 4ac))/(2a )

Nature of roots is determined by the value of the discriminant.

Furthermore, equations are generally written in the decreasing order of exponents, and the standard form of the quadratic equation is in the decreasing order of exponents as well.

The quadratic equation written in the standard from can be factored by the method of splitting the middle term.

Converting a quadratic equation to the general form

Mostly a quadratic equation will not be given in the standard form already. So you have to rewrite it in the standard form. Let us take an example to learn how to convert any quadratic equation to the standard form:
Convert this equation into standard form:

3x² - 5 = 4x

First, we want to get all terms on one side, so subtracting 4x from both sides, we get

3x² - 5 - 4x = 0

Now, we need to arrange the terms in decreasing order of their exponents, so we get

3x² - 4x - 5 = 0

Thus the quadratic equation is converted into the standard form. Now you will be able to use the quadratic formula, or the method of splitting the middle term, in order to solve it.

Obtaining the values of a, b and c from a quadratic equation in the standard form:

It is a matter of simple comparison between the standard quadratic equation and the quadratic equation written in the standard form to obtain the values of a, b and c. For example, for the following quadratic equation,

3x² - 4x - 5 = 0

On comparing with standard quadratic equation,

ax² + bx + c = 0

Therefore a = 3, b = -4 and c = -5

Notice that I have taken the sign on the left of each number along with it when taking the value of a, b and c. You have to do this with all quadratic equations.