106 questions · 21 question types identified
A question is this type if and only if it requires solving an equation of the form ax^(2n) + bx^n + c = 0 where n >= 2 is a positive integer (e.g. x^4, x^6), treated as a quadratic in x^n.
Rearrange a formula where the target variable appears once and only one or two simple steps are needed (e.g. squaring, taking a root, or multiplying out a single bracket), such as E = ½mv², A = πr²(x+y), V = ⅓πr²h, s = ½at².
A question is this type if and only if it requires an explicit substitution (e.g. u = x^(1/3), y = 2^x, u = (3x-2)^2) to reduce a non-standard equation into a solvable quadratic or factorable form.
Determine unknown constants in a quadratic (or polynomial) expression from purely algebraic conditions such as a known minimum/stationary point, specific function values, or given roots, with no real-world context or graph of a trajectory involved.
A question is this type if and only if a real-world quantity (profit, area, fencing) is modelled by a quadratic function and the question asks for the maximum or minimum value and the corresponding variable value, using completing the square or differentiation.
A question is this type if and only if it requires solving a quadratic inequality (e.g. 2x^2 - x - 3 > 0) and expressing the solution as a set of values or intervals, possibly after sketching the curve.
A question is this type if and only if it requires solving a quadratic equation and expressing the roots exactly in surd form (e.g. p ± q√r), typically using the quadratic formula.
Solve equations where substituting u = x^(1/2) (or u = √x) reduces the equation to a standard quadratic in u, e.g. 4x - 11x^(1/2) + 6 = 0, x - 6√x + 4 = 0, 2x - 7x^(1/2) + 3 = 0.
A question is this type if and only if it asks to express a quadratic in completed square form a(x+p)^2 + q, and/or use that form to state the vertex, minimum/maximum value, or solve the equation.
A question is this type if and only if it requires solving an equation where the unknown appears in a denominator (e.g. (3x+1)/(2x) = 4), leading to a linear or simple quadratic after clearing fractions.
Rearrange a formula to make a specified variable the subject where the variable appears more than once, requiring collecting terms and factorising (e.g. y + 5 = x(y + 2), 3(a+4) = ac + 5f, 4h + 5 = 9a - ha²).
A question is this type if and only if it presents a polynomial identity (e.g. 5x^2 + px - 8 ≡ q(x-1)^2 + r) and asks for the values of the unknown constants by expanding and comparing coefficients.
Determine unknown constants in a quadratic model arising from a real-world context (e.g. trajectory of a ball, exchange rate model) where conditions are given via a described physical situation or graph of a trajectory.
Solve equations where the variable appears with negative or reciprocal exponents (e.g. 1/x², 1/x⁴, x^(1/3), x^(2/3)) that form a quadratic structure upon substitution.
Rearrange a formula where the target variable appears once but is embedded in nested operations requiring multiple steps to isolate, such as T = 2π√(l/g), c = √((a+b)/2), a = (√y - 5)/c, or V = ⅓πr²√(l²-r²).
A question is this type if and only if a geometric context (triangle, trapezium, rectangle, L-shape, playground) is used to set up a quadratic equation or expression for area, which is then solved or optimised.
A question is this type if and only if a physical trajectory (ball, arrow, stone) is modelled by a quadratic function H(x) or h(t), and the question asks to find maximum height, range, or interpret constants of the model.
A question is this type if and only if finding the intersection points of two curves (e.g. a polynomial and a line) requires setting up and solving a quadratic or disguised quadratic equation.
A question is this type if and only if it involves similar shapes or geometric figures with surd-valued dimensions, requiring multiplication or simplification of surds to find an unknown length.
A question is this type if and only if it asks about the range of a quadratic function, conditions on constants for real/repeated roots, or properties of roots (e.g. discriminant conditions, sum and product of roots).
Solve equations where substituting u = x^(1/4) reduces the equation to a standard quadratic in u, e.g. 3x^(1/2) - 8x^(1/4) + 4 = 0, 2y^(1/2) - 7y^(1/4) + 3 = 0.