Higher Maths · Relationships and Calculus
Quadratic Theory | Higher Maths
Quadratic Theory in Higher Maths goes beyond solving equations: you need to use the discriminant to classify roots, link the graph to its equation, and solve quadratic inequalities. Expect a 4–6 mark question on this topic in most Higher exams.
Quadratic theory essentials
Quadratic formula: x = (−b ± √(b2 − 4ac)) / (2a)
Discriminant: Δ = b2 − 4ac
If Δ > 0: two real distinct roots
If Δ = 0: two real equal roots (one repeated)
If Δ < 0: no real roots
Worked example
Worked example — Equal roots
Problem: Find the value(s) of k such that the equation x2 + (k + 2)x + 9 = 0 has equal roots.
- For equal roots, the discriminant must be zero.Δ = (k + 2)2 − 4(1)(9) = 0
- Expand and solve.(k + 2)2 = 36 ⇒ k + 2 = ±6
- State both values.k = 4 or k = −8
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Find the discriminant of 2x2 − 5x + 1.
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Δ = 25 − 8 = 17 (two distinct real roots) - For what values of m does mx2 + 4x + m = 0 have no real roots?
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Δ < 0 ⇒ 16 − 4m2 < 0 ⇒ m2 > 4 ⇒ m > 2 or m < −2 - Solve x2 − 2x − 8 < 0.
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(x − 4)(x + 2) < 0 ⇒ −2 < x < 4 - Show that x2 + 6x + 11 has no real roots.
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Δ = 36 − 44 = −8 < 0. No real roots. - Find the range of values of p for which px2 + (p + 2)x + 1 = 0 has real roots.
Show answer
Δ ≥ 0 ⇒ (p + 2)2 − 4p ≥ 0 ⇒ p2 + 4 ≥ 0, true for all real p. So p ∈ ℝ, p ≠ 0.
Common mistakes
Common mistakes & how to avoid them
- Forgetting that Δ = 0 means one repeated root, not zero roots.
- Treating "equal roots" the same as "no roots" — they correspond to different signs of the discriminant.
- Solving quadratic inequalities by simply taking the same answer as the equation: you must consider where the parabola is above or below the x-axis.
Frequently asked questions
Is the quadratic formula on the SQA formula sheet?
The quadratic formula is on the Higher formula sheet, but the discriminant b2 − 4ac is not separately listed — you derive it from the formula.
Can the discriminant be used for any quadratic?
Yes — for any equation in the form ax2 + bx + c = 0, the discriminant tells you the nature of the roots without solving.
What is the difference between roots and intercepts?
Roots are solutions of the equation f(x) = 0. They correspond to the x-intercepts of the graph of y = f(x).
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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