Sign in

Principle deep dive

Vieta — sum and product of roots

If α, β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. Never solve; use structure. Cross-chapter into Complex Numbers, M&D, Properties of Triangle, Trig Identities, and Sequence & Series — and 43% HARD overall, third-toughest principle in the bank.

questions in the bank
49
tagged HARD
43%
chapter spread
7
worked examples below
4

When to reach for it

The question asks for a symmetric function of roots — α + β, αβ, α² + β², (α − β)² — without naming them individually.

Why this principle matters

If α, β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. That's the whole principle. The skill is recognising that any symmetric function of the roots can be written in terms of these two — and that you should NEVER solve for α, β if you're only asked about a symmetric combination.

Vieta is the highest-disguise principle in the bank. NDA dresses it as: 'tan α, tan β are roots of x² − 6x + 8 = 0, find cos(2α + 2β)'. The cosine throws students off; they want to solve the quadratic. But α + β is right there in front of them — that's all they need.

The recipe: extract α + β and αβ from coefficients. Convert the target expression into those. Compute. Three steps, every time.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Quadratic EquationsEASY
Let α\alpha and β\beta (α>β\alpha>\beta) be the roots of the equation x2−8x+q=0x^2-8x+q=0. If α2−β2=16\alpha^2-\beta^2=16, then what is the value of qq?

[Q20 · Apr · 2022]

Example 2Complex NumbersMODERATE
If 2−i32-i\sqrt{3} where i=−1i=\sqrt{-1} is a root of the equation x2+ax+b=0x^{2}+ax+b=0, then what is the value of (a+b)(a+b)?

[Q11 · Apr · 2023]

Example 3Trigonometric IdentitiesMODERATE
If tan⁡α\tan\alpha and tan⁡β\tan\beta are the roots of the equation x2−6x+8=0x^2-6x+8=0, then what is the value of cos⁡(2α+2β)\cos(2\alpha+2\beta)?

[Q48 · Apr · 2024]

Example 4Matrices & DeterminantsHARD
If α\alpha and β\beta are the roots of the equation 1+x+x2=01 + x + x^2 = 0, then the matrix product (1βαα)(αβ1β)\begin{pmatrix}1 & \beta \\ \alpha & \alpha\end{pmatrix}\begin{pmatrix}\alpha & \beta \\ 1 & \beta\end{pmatrix} is equal to

[Q8 · Sep · 2017]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • Three roots: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a

    Cubic case. ax³ + bx² + cx + d = 0. Useful for Roots-of-unity questions.

  • α² + β² = (α + β)² − 2αβ

    Standard substitution. Memorise: also (α − β)² = (α + β)² − 4αβ.

  • sin θ and cos θ as roots

    Use sin²θ + cos²θ = 1 to get b² = a² + 2ac (a classic NDA trick).

  • Reciprocal roots / 1/α + 1/β

    (1/α) + (1/β) = (α + β)/(αβ). Simplifies many cross-chapter questions.

Drill every vieta — sum and product of roots question

49 questions from the bank — paginated, with cart and Word-export support.

Drill the 49 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.