Lens 07 of 9
The boundary
Look at the edge case where the answer changes character: D = 0, a = 0, equal ratios, a degenerate figure.
Many questions ask for a range, and a range is decided at its edge. Real roots meet no real roots exactly where D = 0. Two lines stop crossing exactly where their direction ratios become equal. Find the boundary, test one value on each side, and the whole range is settled.
The clues that call for it
- Quadratic Equations
'For which k are the roots real, or equal?'
D = 0 is the boundary: k² − 4 = 0 → k = ±2.
- Pair of Linear Equations in Two Variables
'For which k do the lines have no solution?'
Parallel is the boundary: a₁/a₂ = b₁/b₂.
Worked example
For which k does x² + kx + 1 = 0 have real roots?
- 1
Push to the boundary
Real roots meet complex roots exactly where the discriminant vanishes.
k² − 4 = 0 → k = ±2
- 2
Test inside
At k = 0 the equation is x² + 1 = 0: no real roots.
- 3
Test outside
At k = 3 the discriminant is 5 > 0: real roots.
- 4
Answer
The boundary separates the two regimes, giving |k| ≥ 2.
|k| ≥ 2
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