CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations
A quadratic polynomial of the form ax² + bx + c, where a ≠ 0 and a, b, c are real numbers, is called a quadratic equation
when ax² + bx + c = 0.
Here a and b are the coefficients of x² and x respectively and ‘c’ is a constant term.
Any value is a solution of a quadratic equation if and only if it satisfies the quadratic equation.
Quadratic formula: The roots, i.e., α and β of a quadratic equation ax² + bx + c = 0 are given
by \(\frac { -b\pm \sqrt { D } }{ 2a } \) or \(\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a } \) provided b² – 4ac ≥ 0.
Here, the value b² – 4ac is known as the discriminant and is generally denoted by D. ‘D’ helps us to determine the nature of roots for a given quadratic equation. Thus D = b² – 4ac.
The rules are:
- If D = 0 ⇒ The roots are Real and Equal.
- If D > 0 ⇒ The two roots are Real and Unequal.
- If D < 0 ⇒ No Real roots exist.
If α and β are the roots of the quadratic equation, then Quadratic equation is x² – (α + β) x + αβ = 0 Or x² – (sum of roots) x + product of roots = 0
where, Sum of roots (α + β) = \(\frac { -coefficient\quad of\quad x }{ coefficient\quad of\quad { x }^{ 2 } } =\frac { -b }{ a } \)
Product of roots (α x β) = \(\frac { coefficient\quad term }{ coefficient\quad of\quad { x }^{ 2 } } =\frac { c }{ a } \)
