1. A quadratic has discriminant zero. How many distinct real roots?
- Two
- One
- None
Check the answer and reasoning
One
The graph touches the axis at a repeated root.
MATH / AS level
Quadratics, functions and coordinate geometry
Completing the square exposes a quadratic’s turning point. The discriminant counts real roots and links algebra with intersections of a line and curve.
Learning objectives
- Explain quadratics, functions and coordinate geometry using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Completing the square exposes a quadratic’s turning point. The discriminant counts real roots and links algebra with intersections of a line and curve.
Domains control inverses and composite functions. Circle equations and line gradients allow geometric constraints to become simultaneous equations; extraneous roots still require checking.
Equations and relationships
Positive, zero and negative D give two, one and no distinct real roots.
Worked example
Find the minimum of x² − 6x + 11.
Follow the solution
Complete the square: (x−3)² + 2, whose minimum occurs at x = 3.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.
Check your understanding
2. Find the minimum of x² − 6x + 11.
- 3
- 11
- 2
Check the answer and reasoning
2
Complete the square: (x−3)² + 2, whose minimum occurs at x = 3.
3. For f(x)=x², what domain makes inverse √x valid?
- x ≥ 0
- All real x
- x ≠ 0
Check the answer and reasoning
x ≥ 0
Restricting to nonnegative inputs makes squaring one-to-one for the stated inverse.