Quadratics — AS & A Level Mathematics (9709) questions by topic
Determining which statement correctly describes a line and curve's intersection, solving a disguised sixth-degree equation, finding a range of constants for which a quadratic stays above a fixed value, and expressing a quadratic in completed-square form with an unknown constant are typical of the 15 quadratics questions from 2023 to 2025, worth 3 to 10 marks.
Find, express, solve, determine and show are the top command words, with show requiring every algebraic step laid out rather than a calculator-only answer. Because completing the square with a leading coefficient is a frequent source of dropped factors, submit your working for instant marking, so a coefficient left unmultiplied through a bracket is caught the instant it happens rather than after the final constant is wrong.
What examiners look for
- A recurring criticism is forgetting to factorise the leading coefficient out of the first two terms before completing the square, which produces an incorrect coefficient structure.
- Examiners note candidates often forgot to multiply the completing-the-square correction back by that same leading factor when combining it with the constant term, giving a wrong value for b.
- The reports flag that most candidates who reached a part (a) result did not go on to exploit it, instead restarting with a fresh substitution and the quadratic formula, or factorising from scratch.
- A recurring criticism is solving a disguised quadratic entirely by calculator with no substitution or factorisation shown, which the mark scheme scores as zero out of three since it requires supported working.
- Strong answers, according to the reports, factorised out the leading 3 first, then completed the square inside the bracket, and verified the result expanded back to the original expression to catch any arithmetic slip on the linear coefficient.
May–June 2025, Paper 11
The equation of a curve is 2x^2 − kxy + 2 = 0 and the equation of a line is y = px + 3, where k and p are constants. Given that k = 2 and p = 11, find the coordinates of the…
Answer this question and get it marked →May–June 2025, Paper 15
Use completing the square to find the exact solutions of the equation 4x^2 − 4x − 1 = 0. [2]
Answer this question and get it marked →May–June 2024, Paper 11
Express 3y^2 - 12y - 15 in the form 3(y + a)^2 + b, where a and b are constants.
Answer this question and get it marked →October–November 2024, Paper 11
Show that the curve with equation x² − 3xy − 40 = 0 and the line with equation 3x + y + k = 0 meet for all values of the constant k.
Answer this question and get it marked →October–November 2024, Paper 12
By expressing −2x² + 8x + 11 in the form −a(x − b)² + c, where a, b and c are positive integers, find the coordinates of the vertex of the graph with equation y = −2x² + 8x + 11.
Answer this question and get it marked →The equation of a curve is y = ½k²x² − 2kx + 2 and the equation of a line is y = kx + p, where k and p are constants with 0 < k < 1. It is given that one of the points of…
Answer this question and get it marked →October–November 2024, Paper 13
Express 3x² − 12x + 14 in the form 3(x + a)² + b, where a and b are constants to be found.
Answer this question and get it marked →May–June 2023, Paper 12
Express 4x² − 24x + p in the form a(x + b)² + c, where a and b are integers and c is to be given in terms of the constant p.
Answer this question and get it marked →Solve the equation 8x⁶ + 215x³ − 27 = 0.
Answer this question and get it marked →May–June 2023, Paper 13
The function f is defined for x ∈ ℝ by f(x) = x² − 6x + c, where c is a constant. It is given that f(x) 2 for all values of x. Find the set of possible values of c.
Answer this question and get it marked →October–November 2023, Paper 11
A line has equation y = 2cx + 3 and a curve has equation y = cx² + 3x − c, where c is a constant. Showing all necessary working, determine which of the following statements is…
Answer this question and get it marked →The curves with equations y = 2(2x − 3)⁴ and y = (2x − 3)² + 1 meet at points A and B. By using the substitution u = 2x − 3 find, by calculation, the coordinates of A and B.
Answer this question and get it marked →Express 4x² − 12x + 13 in the form (2x + a)² + b, where a and b are constants.
Answer this question and get it marked →October–November 2023, Paper 12
The equation of a curve is y = x² − 8x + 5. Find the coordinates of the minimum point of the curve.
Answer this question and get it marked →October–November 2023, Paper 13
A line has equation y = 6x − c and a curve has equation y = cx² + 2x − 3, where c is a constant. The line is a tangent to the curve at point P. Find the possible values of c and…
Answer this question and get it marked →Other AS & A Level Mathematics topics
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