Let us recall the concepts in Quadratic equation:
A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), \(c\) are real numbers, \(a\neq 0\). The degree of the quadratic equation is \(2\).
Important!
The equation \(ax^2 + bx + c = 0\) is called the standard form of a quadratic equation.
Example:
Check whether the equation \((x - 3)^2 + 2 = 3x - 4\) is a quadratic or not.
Solution:
\((x - 3)^2 + 2 = 3x - 4\)
\(\Rightarrow x^2 - 6x + 9 + 2 = 3x - 4\)
\(\Rightarrow x^2 - 6x + 11 -3x + 4 = 0\)
\(\Rightarrow x^2 - 9x + 15 = 0\)
It is of the form \(ax^2 + bx + c = 0\).
Therefore, the given equation is a quadratic equation.
2. Solving the quadratic equation can be done in two different way:
- Solving by factorization.
- Solving by quadratic formula.
Solving by Factorization: If we can factorise \(ax^2+ bx + c\), \(a \neq 0\), into a product of two linear factors, then the roots of the quadratic equation \(ax^2+ bx + c = 0\) can be found by equating each factor to zero.
Solving by Quadratic Formula: The formula for finding roots of the quadratic roots
\(ax^2+bx+c=0\) is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
3. Nature of Roots: A quadratic equation \(ax^2+ bx + c = 0\) has
| Discrimnent | Type of Roots |
| \(b^2-4ac > 0\) | Real and Distinct |
|
\(b^2-4ac = 0\)
|
Real and Equal |
| \(b^2-4ac < 0\) | No Real Roots |