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 $$b^2– 4ac < 0$$.