General form of a Quadratic equation

Standard form

$ax^2+bx+c \,=\, 0$

Introduction

Let $a$, $b$ and $c$ denote three constants, and they represent the real numbers. The literal $x$ represents a variable but its value is unknown here. However, it can be evaluated by solving the following equation.

$ax^2+bx+c \,=\, 0$

The $a$ times $x$ square plus $b$ times $x$ plus $c$ equals to zero is called the standard form of a quadratic equation. It is also called the general form of a quadratic equation.

The quadratic equation in standard form can also be written as follows.

$\implies$ $a \times x^2+b \times x^1+c \times 1 \,=\, 0$

$\,\,\,\therefore\,\,\,\,\,\,$ $a \times x^2+b \times x^1+c \times x^0 \,=\, 0$

Three important factors can be observed from the quadratic equation in general form.

1. The sum of three terms is equal to zero.
2. Each term is a product of a constant and a variable with an exponent.
3. The exponents of a variable in all terms are different and the terms in the equation are written in such a way that the exponents of a variable are in descending order.

Now, it is time to learn more about the constants $a$, $b$ and $c$ in the standard form of a quadratic equation.

1. The constant $a$ is multiplied by the square of $x$ in the first term on the left hand side of the equation. So, The constant $a$ is called the coefficient of $x$ square.
2. The constant $b$ is multiplied by the variable $x$ in the second term. So, The constant $b$ is called the coefficient of variable $x$.
3. The constant $c$ is called the constant term.

Condition

Assume that the value of the coefficient of $x$ square is equal to zero.

$\implies$ $0 \times x^2+bx+c \,=\, 0$

$\implies$ $0+bx+c \,=\, 0$

$\,\,\,\therefore\,\,\,\,\,\,$ $bx+c \,=\, 0$

There is no variable in square form in the equation. Now, the equation is not a quadratic equation and it is a linear equation in one variable. It clears that the coefficient of square of $x$ represents the real numbers but its value should not be equal to zero. Therefore, $a \ne 0$.

Other forms

There are three important cases to learn more about the standard form of a quadratic equation.

b = 0

Consider a case in such a way that the coefficient of variable $x$ is zero in the quadratic equation.

$\implies$ $ax^2+0 \times x+c \,=\, 0$

$\implies$ $ax^2+0+c \,=\, 0$

$\,\,\,\therefore\,\,\,\,\,\,$ $ax^2+c \,=\, 0$

c = 0

Now, let’s assume a case in such a way that the constant is equal to zero.

$\implies$ $ax^2+bx+0 \,=\, 0$

$\,\,\,\therefore\,\,\,\,\,\,$ $ax^2+bx \,=\, 0$

b = 0 & c = 0

Consider a case in such a way that the coefficient of variable $x$ and the constant term are equal to zero.

$\implies$ $ax^2+0 \times x+0 \,=\, 0$

$\implies$ $ax^2+0+0 \,=\, 0$

$\,\,\,\therefore\,\,\,\,\,\,$ $ax^2 \,=\, 0$

The quadratic equation can be expressed in standard form as written in the above three cases, but it is not possible to express a quadratic equation in general form when the coefficient of $x$ square is zero.

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