What is Quadratic Equation in Standard Form

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.