Roots and Graphs of Quadratic Equations
The roots of a quadratic equation and the x-intercepts of its graph are the same thing. Understanding this connection gives a powerful visual way to interpret the discriminant and the nature of roots.
The graph of y = ax² + bx + c is a parabola. Where this parabola crosses the x-axis (if at all) tells us the roots of ax² + bx + c = 0.
This topic connects two chapters of Class 10 NCERT — Polynomials (graphs) and Quadratic Equations (roots and discriminant).
What is Roots and Graphs of Quadratic Equations?
Definition: The roots (or zeros) of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy the equation. Graphically, these are the x-coordinates of the points where the parabola y = ax² + bx + c crosses the x-axis.
The Discriminant D = b² − 4ac determines the nature of roots:
- D > 0: Two distinct real roots — parabola cuts the x-axis at two points.
- D = 0: Two equal real roots (repeated root) — parabola touches the x-axis at exactly one point.
- D < 0: No real roots — parabola does not touch the x-axis at all.
Roots and Graphs of Quadratic Equations Formula
Quadratic Formula:
x = (−b ± √D) / 2a, where D = b² − 4ac
Vertex of the parabola:
Vertex = (−b/2a, −D/4a)
Sum and Product of roots:
- Sum of roots (α + β) = −b/a
- Product of roots (αβ) = c/a
Types and Properties
Three cases of roots vs graphs:
| Discriminant | Nature of Roots | Graph |
|---|---|---|
| D > 0 | Two distinct real roots α, β | Parabola cuts x-axis at two points (α, 0) and (β, 0) |
| D = 0 | One repeated real root α = β = −b/2a | Parabola touches x-axis at one point (−b/2a, 0) |
| D < 0 | No real roots (complex roots) | Parabola lies entirely above (a > 0) or below (a < 0) the x-axis |
Direction of parabola:
- If a > 0: parabola opens upward (U-shape) — minimum at vertex.
- If a < 0: parabola opens downward (∩-shape) — maximum at vertex.
Solved Examples
Example 1: Two Distinct Roots (D > 0)
Problem: Find the roots of x² − 5x + 6 = 0 and describe the graph.
Solution:
Discriminant: D = 25 − 24 = 1 > 0 → two distinct real roots.
Roots:
- x = (5 ± 1)/2 → x = 3 or x = 2
Graph: y = x² − 5x + 6 is an upward parabola (a = 1 > 0) that crosses the x-axis at (2, 0) and (3, 0).
Vertex: x = 5/2 = 2.5, y = (2.5)² − 5(2.5) + 6 = 6.25 − 12.5 + 6 = −0.25. Vertex = (2.5, −0.25).
Answer: Roots are 2 and 3. The parabola dips below the x-axis between x = 2 and x = 3.
Example 2: Equal Roots (D = 0)
Problem: Find the roots of x² − 6x + 9 = 0 and describe the graph.
Solution:
Discriminant: D = 36 − 36 = 0 → equal (repeated) roots.
Root:
- x = 6/2 = 3 (both roots are 3)
Graph: y = x² − 6x + 9 = (x − 3)² is an upward parabola that just touches the x-axis at (3, 0). The vertex is the point of tangency.
Answer: Both roots are 3. The parabola touches the x-axis at exactly one point.
Example 3: No Real Roots (D < 0)
Problem: Determine the nature of roots of x² + 2x + 5 = 0 and describe the graph.
Solution:
Discriminant: D = 4 − 20 = −16 < 0 → no real roots.
Graph: y = x² + 2x + 5. Since a = 1 > 0, it opens upward. Vertex: x = −1, y = 1 − 2 + 5 = 4. Vertex = (−1, 4).
The entire parabola lies above the x-axis (minimum y-value is 4 > 0).
Answer: No real roots. The parabola does not cross or touch the x-axis.
Example 4: Finding k for Equal Roots
Problem: Find the value of k for which 2x² + kx + 8 = 0 has equal roots.
Solution:
For equal roots, D = 0:
- k² − 4(2)(8) = 0
- k² = 64
- k = ±8
Answer: k = 8 or −8.
Example 5: Downward Parabola
Problem: Analyse the roots and graph of −x² + 4x − 3 = 0.
Solution:
Discriminant: D = 16 − 12 = 4 > 0 → two distinct roots.
Roots:
- x = (−4 ± 2)/(−2) → x = (−4+2)/(−2) = 1 or x = (−4−2)/(−2) = 3
Graph: y = −x² + 4x − 3. Since a = −1 < 0, it opens downward. Vertex: x = −4/(−2) = 2, y = −4 + 8 − 3 = 1. Vertex = (2, 1).
The parabola crosses the x-axis at (1, 0) and (3, 0), with the vertex above the x-axis.
Answer: Roots are 1 and 3. Downward parabola with vertex (2, 1).
Example 6: Graph Reading — Finding the Equation
Problem: A parabola crosses the x-axis at x = −1 and x = 4, and passes through (0, −4). Find the quadratic equation.
Solution:
Since roots are −1 and 4: y = a(x + 1)(x − 4).
Passes through (0, −4): −4 = a(1)(−4) → a = 1.
Equation: y = (x + 1)(x − 4) = x² − 3x − 4.
Answer: The equation is x² − 3x − 4 = 0.
Example 7: Range of k for No Real Roots
Problem: For what values of k does kx² + 2x + 1 = 0 have no real roots?
Solution:
For no real roots, D < 0 (and k ≠ 0):
- 4 − 4k < 0
- 4 < 4k
- k > 1
Answer: No real roots when k > 1.
Example 8: Connecting Sum/Product of Roots to Graph
Problem: For y = 2x² − 7x + 3, find the roots, their sum, product, and verify from the graph.
Solution:
Roots: D = 49 − 24 = 25. x = (7 ± 5)/4 → x = 3 or x = 1/2.
Sum of roots: 3 + 1/2 = 7/2 = −b/a = 7/2 ✓
Product of roots: 3 × 1/2 = 3/2 = c/a = 3/2 ✓
Graph: Upward parabola crossing x-axis at x = 0.5 and x = 3. Vertex at x = 7/4 = 1.75.
Answer: Roots are 1/2 and 3. Sum = 7/2, Product = 3/2.
Real-World Applications
The connection between roots and graphs is used in:
- Physics: The trajectory of a projectile is a parabola; the roots tell when the object hits the ground.
- Economics: Profit = Revenue − Cost; the roots of the profit equation give break-even points.
- Engineering: Parabolic arches and cables; the vertex gives the maximum height.
- Data Analysis: Fitting a quadratic to data and identifying when the dependent variable equals zero.
- Signal Processing: Filters and resonance conditions are determined by the roots of quadratic characteristic equations.
Key Points to Remember
- The roots of ax² + bx + c = 0 are the x-intercepts of y = ax² + bx + c.
- D > 0: two x-intercepts; D = 0: one x-intercept (vertex on x-axis); D < 0: no x-intercepts.
- If a > 0, parabola opens upward; if a < 0, parabola opens downward.
- The vertex is at (−b/2a, −D/4a).
- The axis of symmetry is x = −b/2a (midpoint of the two roots).
- Sum of roots = −b/a; Product of roots = c/a.
- For equal roots, set D = b² − 4ac = 0 and solve for the unknown parameter.
- A parabola that opens upward with D < 0 lies entirely above the x-axis (all y > 0).
- The sign of 'a' determines the direction; the discriminant determines the position relative to the x-axis.
- If given the x-intercepts α and β, the equation can be written as a(x − α)(x − β) = 0.
Practice Problems
- Determine the nature of roots and draw a rough sketch: x² − 4x + 4 = 0.
- For what value of k does x² + kx + 9 = 0 have equal roots?
- Find the roots and vertex of y = −2x² + 8x − 6. Sketch the parabola.
- A parabola has roots at x = 2 and x = 5, and passes through (3, −2). Find the equation.
- Prove that x² + 3x + 5 = 0 has no real roots. What does the graph look like?
- Find the range of p for which px² − 4x + 1 = 0 has two distinct real roots.
Frequently Asked Questions
Q1. What is the visual meaning of the discriminant?
The discriminant tells how many times the parabola crosses the x-axis. D > 0 means two crossings, D = 0 means it just touches, and D < 0 means it never reaches the x-axis.
Q2. What does the vertex represent?
The vertex is the lowest point (if a > 0) or highest point (if a < 0) of the parabola. It lies on the axis of symmetry x = −b/2a.
Q3. How are the roots related to the axis of symmetry?
The two roots are equidistant from the axis of symmetry. If the roots are α and β, the axis of symmetry is x = (α + β)/2 = −b/2a.
Q4. Can the graph give exact roots?
Only if the roots are integers or simple fractions that can be read precisely from the graph. For irrational roots, the graph gives approximate values; the quadratic formula gives exact values.
Q5. How do you find a quadratic equation from its graph?
If the x-intercepts are α and β, the equation is y = a(x − α)(x − β). Use one more point on the graph to find the value of a.
Q6. What happens if a = 0?
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The graph is a straight line, not a parabola.
Related Topics
- Discriminant of Quadratic Equation
- Nature of Roots of Quadratic Equation
- Graph of Quadratic Polynomial
- Quadratic Formula
- Quadratic Equations
- Standard Form of Quadratic Equation
- Solving Quadratic Equations by Factorisation
- Factorisation of Quadratic Expressions
- Completing the Square Method
- Sum and Product of Roots
- Word Problems on Quadratic Equations
- Applications of Quadratic Equations
