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The graph of is a parabola that touches the -axis at exactly one point without crossing it. What does this say about the zeroes of ?

Class 10PolynomialsAlgebra — Polynomials
  • AIt has two distinct zeroes
  • BIt has no real zeroes
  • CIt has one zero, and that zero is repeated
  • DIt has three zeroes

Step-by-Step Solution

One touching point means one -intercept, so one distinct real zero. Writing the quadratic as a − α) − β), a touch happens exactly when α = β, so the zero is repeated (discriminant 0). Two distinct zeroes would need the curve to cross the axis at two separate points, and 'no real zeroes' would mean it never meets the axis at all.
Answer: C) It has one zero, and that zero is repeated

Key idea: Geometric meaning of zeros

Zeros of are exactly the **-coordinates where the graph meets the -axis**.

- **Linear** = ax + : graph is a straight line, crosses -axis ONCE → exactly 1 zero.
- **Quadratic** : graph is a parabola. Possible cases:
- Crosses -axis at 2 points → 2 distinct real zeros.
- Touches -axis at 1 point → 1 repeated zero (vertex on -axis).
- Does not meet -axis → no real zeros.
- **Cubic** + …: shape is a curve that crosses 1, 2, or 3 times → 1, 2, or 3 real zeros.

So 'find the zeros' is the algebraic version of 'find where the graph crosses the -axis'.

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