Discriminant and the nature of roots
The discriminant D = b² − 4ac decides the roots. D > 0: two distinct real roots. D = 0: two equal roots. D < 0: no real roots.
The three cases
| Discriminant | Nature of roots | Graph of y = ax² + bx + c |
|---|---|---|
| D > 0 | Two distinct real roots | Cuts the x-axis at two points |
| D = 0 | Two equal real roots, x = −b/(2a) | Touches the x-axis at one point |
| D < 0 | No real roots | Does not meet the x-axis |
Example 1: find the nature of the roots
Find the nature of the roots of (i) 2x² − 5x + 3 = 0 (ii) 9x² − 12x + 4 = 0 (iii) x² + 2x + 4 = 0.
- (i) D = (−5)² − 4(2)(3) = 25 − 24 = 1 > 0: two distinct real roots.
- (ii) D = (−12)² − 4(9)(4) = 144 − 144 = 0: two equal real roots, each 12/18 = 2/3.
- (iii) D = 2² − 4(1)(4) = 4 − 16 = −12 < 0: no real roots.
(i) distinct real (ii) equal real, x = 2/3 (iii) no real roots
Example 2: find k for equal roots
Find k so that kx² − 6x + 1 = 0 has two equal roots. Find the roots.
- For equal roots, D = 0. Here a = k, b = −6, c = 1.
- (−6)² − 4(k)(1) = 0, so 36 − 4k = 0 and k = 9.
- Each root = −b/(2a) = 6/18 = 1/3.
- Check: 9x² − 6x + 1 = (3x − 1)², which is 0 when x = 1/3.
k = 9, and both roots are 1/3.
Example 3: is the design possible?
Can a rectangular rangoli border be made with a perimeter of 40 cm and an area of 120 cm²?
- Perimeter 40 cm means length + breadth = 20 cm.
- Let the breadth be x cm. Then the length is (20 − x) cm.
- Area: x(20 − x) = 120, so x² − 20x + 120 = 0.
- D = (−20)² − 4(1)(120) = 400 − 480 = −80
- D < 0, so there is no real value of x.
No. Such a rectangle is not possible.
Which condition to use in 'find k' questions
- Two equal roots ('real and equal'): D = 0.
- Real roots: D ≥ 0.
- Distinct real roots: D > 0.
- No real roots: D < 0.
- If the x² coefficient contains k, also make sure it is not zero.
A carpenter may need a frame with a given perimeter and area. Before any wood is cut, the discriminant tells whether such a frame can exist. If D < 0, no real length works.
1 marks: “Find the nature of the roots of x² − 4x + 5 = 0.” D = 16 − 20 = −4 < 0, so there are no real roots.
2 marks: “Find the values of k for which x² − 2kx + 9 = 0 has equal roots.” D = 4k² − 36 = 0, so k² = 9 and k = ±3.
3 marks: “Find p for which (p − 3)x² + 6x + 9 = 0 has two equal real roots, and find the roots.” D = 36 − 36(p − 3) = 0, so p − 3 = 1 and p = 4. The equation x² + 6x + 9 = 0 gives x = −3, −3.
Positive D: two paths. Zero D: one meeting point. Negative D: no real way.