The big picture
One line covers it all. Every question is a substitution, a bracket, or a coefficient.
| Idea | Remember | Example |
|---|---|---|
| The zero | An input where the value is 0. | x = 8 for −5x² + 40x |
| Brackets | (x − p)(x − q) has zeros p and q. | (x − 2)(x − 3) gives 2, 3 |
| The curve | The zeros are where the curve meets the x-axis. | x² + 1 never meets the x-axis |
| The sum | If ax² + bx + c has zeros, their sum is −b/a. | 2x² − 12x + 10: sum 6 |
| The product | If ax² + bx + c has zeros, their product is c/a. | 2x² − 12x + 10: product 5 |
| Backwards | Zeros with sum S and product P give x² − Sx + P. | 4, 5 give x² − 9x + 20 |
| The signs | If zeros exist, a positive product means they share a side. A negative product means they sit on opposite sides. | x² + 7x + 12: both negative |
| Values from S and P | α² + β² = S² − 2P. | S = 4, P = 1 gives 14 |
| The cubic | α + β + γ = −b/a. αβ + βγ + γα = c/a. αβγ = −d/a. | zeros 2, −1, 3/2: sum 5/2 |
Open Chapter 2, Polynomials, in your NCERT book. Read its solved examples. They use the methods you have just learned.