Here you will learn quadratic equation concepts and what is quadratic equation in standard form.

Let’s begin –

## Quadratic Equation

If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.

For example, \(x^2 + 2x – 8\) = 0, \(x^2 – 5x + 6\) = 0 are quadratic equations.

The general form of quadratic equation is \(ax^2 + bx + c\) = 0. where a, b, c \(\in\) R and a \(\ne\) 0.

## What is Quadratic Equation in Standard Form ?

The equation of the form \(ax^2 + bx + c\) = 0, a \(\ne\) 0 is known as standard equation.

For example, \(5x^2 + 3x + 6\) = 0 is a quadratic equation in the standard form.

## Rule to Determine Whether Equation is Quadratic or Not

1). Write down the given equation in the form f(x) = 0.

2). (a) If f(x) is a polynomial, then observe its degree.

(b) If f(x) is not a polynomial, then first make it poynomial and then observe its degree.

3). If degree of the polynomial is 2, then the given equation is quadratic.

**Example** : Which of the following are quadratic equation ?

(i) \(x^2 -6x – 4\) = 0

(ii) x + 2 = 0

(iii) x + \(1\over x\) = 1, x \(\ne\) 0

(iv) \(x^2\) + \(1\over x\) = 1, x \(\ne\) 0

**Solution** :

(i) p(x) = \(x^2 -6x – 4\) is polynomial with degree 2.

\(\therefore\) \(x^2 -6x – 4\) = 0 is a quadratic equation.

(ii) p(x) = x + 2 is polynomial with degree 1.

\(\therefore\) x + 2 = 0 is not a quadratic equation.

(iii) x + \(1\over x\) = 1 \(\implies\) \(x^2 + 1\over x\) = 1 \(\implies\) \(x^2 +1\) = x \(\implies\) \(x^2 – x + 1\) = 0

Since, here p(x) = \(x^2 – x + 1\) is polynomial with degree 2.

\(\therefore\) \(x^2 – x + 1\) = 0 is a quadratic equation.

(iv) \(x^2\) + \(1\over x\) = 1 \(\implies\) \(x^3 + 1\over x\) = 1 \(\implies\) \(x^3 +1\) = x \(\implies\) \(x^3 – x + 1\) = 0

Here, p(x) = \(x^3 – x + 1\) is polynomial with degree 3.

\(\therefore\) \(x^3 – x + 1\) = 0 is not a quadratic equation.