There will be obstacles. There will be doubters. There will be mistakes. But with hard work, there are no limits. – **Michael Phelps**

In this chapter, you have studied the following points:

1. A quadratic equation in the variable x is of the form \(ax^2+bx+c=0\), where a, b, c are real numbers and a ≠ 0.

2. A real number α is said to be a root of the quadratic equation \(ax^2 + bx + c = 0\), if \(aα^2+ bα + c = 0\). The zeroes of the quadratic polynomial \(ax^2 + bx + c = 0\) and the roots of the quadratic equation \(ax^2+ bx + c\)

3. If we can factorise \(ax^2+bx+c\), \(a \ne 0\) into a product of two linear factors, then the roots of the quadratic equation \(ax^2+bx+c\) can be found by equating each factor to zero.

4. A quadratic equation can also be solved by the method of completing the square.

5. Quadratic formula: The roots of a quadratic equation \(ax^2+bx+c=0\) are given by \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), provided \(b^2-4ac \ge 0\).

6. A quadratic equation \(ax^2 + bx + c = 0\) has

(i) two distinct real roots, if \(b^2 – 4ac > 0\),

(ii) two equal roots (i.e., coincident roots), if \(b^2 – 4ac = 0\), and

(iii) no real roots, if \(b^2– 4ac < 0\).