The bit under the root: The discriminant is Δ = b² − 4ac — the expression under the square root in the quadratic formula. Its sign tells you how many real roots the quadratic has, without solving it.
IB-style question — compute Δ
Find the discriminant of 2x² − 4x + 1.
Step by step
- a = 2, b = −4, c = 1.
- Evaluate.
Final answer
Δ = 8 (positive, so two real roots).
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Sign of Δ → number of roots: Δ > 0 → two distinct real roots (graph cuts the x-axis twice). Δ = 0 → one repeated root (graph touches the x-axis). Δ < 0 → no real roots (graph misses the x-axis).
Δ > 0
- Two distinct roots
- Cuts x-axis twice
Δ = 0
- One repeated root
- Touches x-axis (tangent)
Δ < 0
- No real roots
- Misses the x-axis
You don't need to solve: If a question only asks 'how many solutions' or 'show it has no real roots', just compute Δ and read its sign.
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Tangent = touches at exactly one point: Tangent is just a fancy word for touching at exactly one point — the line grazes the curve instead of cutting through it (picture a ball resting on a flat floor: they meet at a single spot). When you set the line equal to the curve you get a quadratic, and one touching point means one solution — which means the discriminant is Δ = 0.
IB-style question — find the tangent
Find the value of c for which the line y = x + c is tangent to (just touches) the curve y = x².
Step by step
- Set the line equal to the curve, then move everything to one side to get a quadratic = 0.
- One touch means one solution, so the discriminant is 0. Write the formula first.
- Match x² − x − c to ax² + bx + c: here a = 1, b = −1, and the constant term is −c. Now substitute.
- Solve for c.
Final answer
c = −1/4 — at this value the line just touches the parabola at a single point.
Two graphs meet → use the discriminant: Whenever a line and a curve meet a set number of times, set them equal, form a quadratic, and look at Δ = b² − 4ac: Δ = 0 → just touches (tangent), Δ > 0 → crosses at two points, Δ < 0 → never meet.
Turn the root-condition into an equation in the unknown: "Equal roots" → Δ = 0; "two distinct roots" → Δ > 0; "no real roots" → Δ < 0. Write Δ in terms of the unknown and solve the resulting equation or inequality.
IB-style question — equal roots
The equation x² + kx + 9 = 0 has equal roots. Find the positive value of k.
Step by step
- "Equal roots" means one repeated root, so the discriminant is 0. Write the formula first.
- Match x² + kx + 9 to ax² + bx + c: a = 1, b = k, c = 9. Now substitute.
- Solve for k.
Final answer
k = 6 (the positive value).
Read the condition carefully: "Two different solutions" is Δ > 0 (an inequality); "a repeated/equal root" is Δ = 0 (an equation). Match the wording.