Here you will learn quadratic equation concepts and what is quadratic equation in standard form.
Letโs begin โย
Quadratic Equation
If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.
For example, \(x^2 + 2x โ 8\) = 0, \(x^2 โ 5x + 6\) = 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) \(x^2 -6x โ 4\) = 0
(ii) x + 2 = 0
(iii) x + \(1\over x\) = 1,ย x \(\ne\) 0
(iv) \(x^2\) + \(1\over x\) = 1,ย x \(\ne\) 0
Solution :ย
(i) p(x) = \(x^2 -6x โ 4\) is polynomial with degree 2.
\(\therefore\)ย \(x^2 -6x โ 4\) = 0 is a quadratic equation.
(ii) p(x) = x + 2 is polynomial with degree 1.
\(\therefore\)ย 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 \(\implies\) \(x^2 โ x + 1\) = 0
Since, here p(x) = \(x^2 โ x + 1\) is polynomial with degree 2.
\(\therefore\)ย \(x^2 โ x + 1\) = 0 is a quadratic equation.
(iv)ย \(x^2\) + \(1\over x\) = 1 \(\implies\) \(x^3 + 1\over x\) = 1 \(\implies\) \(x^3 +1\) = x \(\implies\) \(x^3 โ x + 1\) = 0
Here, p(x) = \(x^3 โ x + 1\) is polynomial with degree 3.
\(\therefore\)ย \(x^3 โ x + 1\) = 0 is not a quadratic equation.
