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.