How many marks are quadratics worth in 9709 Paper 1?

Quadratics directly accounts for 5-8 marks out of 75 on Paper 1 (7-11%). But quadratic techniques — the discriminant, completing the square, hidden quadratics — appear embedded in 30+ marks of other topic questions across coordinate geometry, functions, and trigonometry. When you account for these, the true contribution is closer to 40% of the paper.

Is completing the square harder than the quadratic formula?

They are different tools for different jobs. The quadratic formula finds roots. Completing the square gives you the vertex form a(x + p)² + q, which you need for finding the range of a function, the turning point of a curve, and converting circle equations. You need both for 9709 — you cannot rely on one alone.

How do I spot a hidden quadratic?

Look for three terms where the power of the first term is twice the power of the second. If you see sin²x alongside sin x, or 4^x alongside 2^x, or x⁴ alongside x² — it's a hidden quadratic. Let u equal the lower-power term and solve the resulting quadratic in u.

Do I need to know the quadratic formula for 9709?

Yes. Completing the square isn't always the quickest method, especially when the coefficients are awkward. The formula is also useful when a question asks you to 'solve' rather than 'express in the form.' Both are in the syllabus and both appear in mark schemes.

What is the most common quadratic mistake in 9709?

Sign errors when completing the square with a negative leading coefficient. Cambridge examiners cite this in their reports every session. The fix: factor out the negative first, then complete the square inside the bracket, and expand your answer back out to verify.

A Level Maths Quadratics: 9709 Pure 1 Guide | ExamPilot

This guide is part of our Complete Cambridge 9709 Pure 1 Revision Guide.

What Cambridge 9709 Quadratics Actually Tests

You can already solve A Level Maths quadratics. Factorising, the quadratic formula, sketching parabolas — you've covered all of that at GCSE. So what makes the Cambridge 9709 version different? The short answer: 9709 doesn't test whether you can solve quadratics. It tests whether you can use quadratic techniques as tools inside other problems. The discriminant proves a line is tangent to a curve. Completing the square is also the technique that rewrites a circle equation into centre-radius form in coordinate geometry — one example of how this quadratic skill recurs across other topics. A trigonometry question that looks nothing like a quadratic turns out to be one in disguise.

What's in scope for Paper 1

Completing the square: express ax² + bx + c in the form a(x + p)² + q
The discriminant b² −4ac and conditions for real, equal, and no real roots
Solving quadratic equations by factorising, completing the square, and the formula
Solving quadratic inequalities
Solving simultaneous equations where one is linear and one is quadratic
Recognising and solving equations that are quadratic in some function of x (hidden quadratics), e.g. x⁴ − 5x² + 4 = 0 or tan²x = 1 + tan x

What's NOT in Pure 1 scope: partial fractions (Pure 3), complex roots (Further Maths). Papers 11, 12, and 13 are simply the three variants of Paper 1 — if you're sitting any of them, the content is identical across all three. The focus is on quadratic techniques as tools applied within other topics.

Completing the Square — The Skill That Unlocks Everything

Completing the square is the single most transferable quadratic technique in Pure 1. The method converts ax² + bx + c into vertex form a(x + p)² + q, which gives you the turning point directly. It appears in quadratics questions, circle equations in coordinate geometry, range of functions, and maximum/minimum problems. Get this right and you unlock marks across half the paper.

Step-by-step completing the square with sign tracking — Completing the square: step-by-step with sign tracking

When written in the form a(x + p)² + q, the vertex form gives the turning point directly at (−p, q). For a monic quadratic (leading coefficient 1), halve the coefficient of x, square it, and adjust. For a non-monic quadratic, factor out the leading coefficient first. This means you can sketch the parabola, find the range of a function, or identify the maximum/minimum without differentiation.

The Discriminant in Action

The discriminant is not a formula to memorise and apply mechanically. It is a decision tool that tells you what kind of solutions an equation has. The three conditions — greater than zero, equal to zero, less than zero — each have different geometric interpretations and appear across at least four different topic areas in 9709.

Three parabolas showing two roots, one root, and no real roots — The three discriminant conditions: two roots, one root, no real roots

In coordinate geometry: 'Show that the line is a tangent to the curve' means substitute, rearrange, and set b² − 4ac = 0 because a tangent touches at exactly one point. In functions: 'Find the range of values of k for which f(x) = k has no solutions' means set f(x) = k, rearrange into a quadratic, and apply b² − 4ac < 0. In 'find the values of k' questions, identify a, b, c in terms of k, set the appropriate discriminant condition, and solve the resulting equation or inequality.

Hidden Quadratics — Spotting the Disguise

Hidden quadratics are the distinctly A-Level concept that GCSE didn't prepare you for. These are equations that don't look like quadratics but become quadratics after a substitution. The trick is recognition: any equation where the powers of the three terms are in the ratio 2 : 1 : 0 can be reduced by substitution. It appears in trigonometry, exponentials, and algebraic questions — often without any hint that there's a quadratic involved.

Flowchart for recognising and solving hidden quadratics — How to spot and solve a hidden quadratic

The substitution checklist: (1) Identify the 'base' expression (e.g. sin x, x²). (2) Check the powers — is the first term the square of the second? (3) Substitute u = [base expression]. (4) Solve the quadratic in u. (5) Convert back to the original variable. (6) Check validity — reject solutions outside the domain. Step 6 is where marks are lost. If u = sin x, then −1 ≤ u ≤ 1.

Quadratic Inequalities — The Sketch Method

The reliable method for quadratic inequalities: (1) Solve the corresponding equation to find the critical values. For x² − 5x + 6 = 0, factorise to get x = 2 and x = 3. (2) Sketch the parabola — positive x² coefficient means it opens upward, negative means downward. (3) Mark the roots on the x-axis. (4) Read the inequality from the graph — shade above the x-axis for 'greater than zero,' shade below for 'less than zero.' For x² − 5x + 6 > 0, the solution is x < 2 or x > 3. For x² − 5x + 6 < 0, the solution is 2 < x < 3.

Parabola with shaded regions for less-than and greater-than inequalities — Solving quadratic inequalities: sketch the parabola, read the solution

How Quadratics Connects to Every Other Topic

Quadratics is the most foundational topic in Pure 1. It doesn't just appear in the 'quadratics' question — it's everywhere. Master completing the square and the discriminant, and you have tools that earn marks in at least four other topics besides quadratics itself.

Hub diagram showing quadratics connected to five other topics — Quadratics is the foundation beneath half the Pure 1 syllabus

The discriminant condition for tangents (b² − 4ac = 0) is tested in coordinate geometry questions almost every session. Completing the square converts general circle equations to centre-radius form. See our 9709 coordinate geometry guide for the full method.

Hidden quadratics appear in trigonometry equations like sin²x + 3sin x + 2 = 0. The substitution technique is identical. See our 9709 trigonometry guide for worked examples.

Vertex form from completing the square gives you the range of quadratic functions directly. See our 9709 functions guide for range and domain methods.

Setting a derivative equal to zero to find stationary points often produces a quadratic to solve. See our 9709 differentiation guide for optimisation problems.

What to Do Next

Master completing the square first — including negative coefficients — because it appears the most frequently and errors here cascade into other topics. Then learn the three discriminant conditions and where each one applies. Build your pattern recognition for hidden quadratics after that. Those three skills unlock marks across half the paper. Paper 1 carries 75 marks out of 250 total for the full A-Level. Quadratic techniques touch the majority of those 75 marks. The time you invest here pays off everywhere.

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Looking for past paper practice? See our 9709 Past Papers by Topic collection.

Key Formulas

Completing the Square

x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + c - \left(\frac{b}{2}\right)^2

Completing the square for a monic quadratic

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Completing the square with a leading coefficient

The vertex form a(x + p)² + q, where p = b/2a and q = c − b²/4a, gives the turning point directly at (−p, q). No calculus needed. This same technique converts general circle equations to centre-radius form in coordinate geometry — complete the square on both x and y separately to find the centre and radius.

Completing the Square with Negative Leading Coefficient

-x^2 + 6x - 5 = -(x^2 - 6x) - 5 = -(x - 3)^2 + 9 - 5 = -(x - 3)^2 + 4

Factor out the negative first, then complete the square inside the bracket

Factor out the negative sign FIRST, then complete the square inside the bracket. The most common error is writing -9 - 5 instead of +9 - 5 when distributing the negative sign. Sign errors with negative leading coefficients are among the most commonly reported mistakes in Cambridge examiner reports. Always distribute the factored-out coefficient back through before moving on, and verify by expanding your answer.

The Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The quadratic formula — use when factorising is not straightforward

Completing the square and the quadratic formula are different tools for different jobs. Completing the square gives you the vertex form, which you need for finding the range of a function, the turning point of a curve, and converting circle equations. The quadratic formula finds roots directly and is faster when the coefficients are awkward. You need both for 9709.

The Discriminant

\Delta = b^2 - 4ac

The discriminant determines the nature of the roots

b^2 - 4ac > 0 \implies \text{two distinct real roots}

Two distinct real roots

b^2 - 4ac = 0 \implies \text{one repeated root (tangent condition)}

One repeated root — the tangent condition in coordinate geometry

b^2 - 4ac < 0 \implies \text{no real roots}

No real roots

The discriminant is not just a formula to memorise. It is a decision tool that appears across at least four topic areas: coordinate geometry (tangent conditions — set b² − 4ac = 0), functions (range problems — set b² − 4ac < 0), quadratics ('find the values of k' questions), and trigonometry (checking validity after hidden quadratic substitution).

Quadratic Inequalities

ax^2 + bx + c > 0 \quad \text{or} \quad ax^2 + bx + c < 0

Solve by finding critical values, then sketch the parabola

Find the critical values (roots) first by solving the corresponding equation. Then sketch the parabola — check whether the coefficient of x² is positive (opens upward) or negative (opens downward). Mark the roots on the x-axis and read the inequality from the graph. The sketch method is more reliable than working algebraically from the factorised form because it makes the solution range visually obvious and avoids the common error of treating the inequality as an equation.

Worked Examples

Completing the Square (Monic)

Easy[2 marks]

Express x² + 6x + 2 in the form (x + p)² + q.

Completing the Square (Negative Leading Coefficient)

Medium[3 marks]

Express -x² + 6x - 5 in the form a(x + p)² + q.

Using the Discriminant to Find k

Medium[4 marks]

Find the values of k for which x² + kx + 9 = 0 has equal roots.

Hidden Quadratic (Trigonometric)

Medium[5 marks]

Solve sin²x + 3sin x + 2 = 0 for 0° ≤ x ≤ 360°.

Quadratic Inequality

Medium[3 marks]

Solve x² − 5x + 6 < 0.

Simultaneous Equations (Quadratic-Linear)

Medium[4 marks]

Solve simultaneously: y = x + 3 and y = x² + x − 1.

Common Mistakes

Sign errors when completing the square, especially with negative leading coefficients

Factor out the negative sign FIRST: −x² + 6x − 5 becomes -(x² − 6x) − 5. Then complete the square inside the bracket. The most common error is writing −9 − 5 instead of +9 − 5 when distributing the negative sign. Cambridge examiner reports note that sign errors when completing the square were widespread, particularly when the coefficient of x² was negative. Always write each step on a new line, distribute the factored-out coefficient back through, and verify by expanding your answer.

Applying the wrong discriminant condition (using > 0 when = 0 is needed)

Tangent, 'touches,' or 'equal roots' all mean b² − 4ac = 0, NOT > 0. Two distinct real roots means > 0. No real roots means < 0. Examiner reports flag this confusion every session, as many candidates stated b² − 4ac > 0 when the question required equal roots (2022). Learn all three conditions as a set, and read the question for the exact wording before selecting the condition.

Not checking hidden quadratic solutions against the domain

After solving a hidden quadratic, always convert back and check validity. If u = sin x, then u must be between −1 and 1 — reject sin x = 3. The examiner report notes that candidates who found both solutions of the quadratic but failed to reject the invalid one lost the final mark (2024). Always apply step 6 of the substitution checklist.

Solving a quadratic inequality as an equation (writing x = ... instead of a range)

A quadratic inequality has a range as its answer, not specific values. If x² − 5x + 6 < 0, the answer is 2 < x < 3, NOT x = 2 or x = 3. Always sketch the parabola and shade the correct region. Cambridge examiners note that a significant number of candidates gave the critical values as the answer rather than the inequality (2023). Check the mark scheme — examiners require the inequality notation, not just the critical values.

Confusing completing the square with solving the equation

Completing the square converts to vertex form a(x + p)² + q. Solving finds roots. When the question asks to 'express in the form a(x + p)² + q,' do NOT set the expression equal to zero. 'Express in the form' and 'solve' are different instructions. Students lose marks by completing the square correctly and then writing x = ... when the question only asked for the form.

Exam Tips

How to Spot a Hidden Quadratic

Hidden quadratics — also known as disguised quadratics — are equations that don't look like quadratics but become quadratics after a substitution. The pattern to recognise: three terms where the power of the first term is twice the power of the second. This is the A-Level concept that GCSE didn't prepare you for. It appears in trigonometry and algebraic questions — often without any hint that there's a quadratic involved.

Common hidden quadratic disguises in 9709

Original Equation	Substitution	Quadratic Form
sin²x + 3sin x + 2 = 0	Let u = sin x	u² + 3u + 2 = 0
2sin²x − sin x − 1 = 0	Let u = sin x	2u² − u − 1 = 0
x⁴ − 5x² + 4 = 0	Let u = x²	u² − 5u + 4 = 0
x + 1/x = 5	Multiply by x, rearrange	x² − 5x + 1 = 0

The Discriminant as a Decision Tool

The discriminant isn't just a quadratics tool. It appears across at least four other topics in 9709. Students who learn it only as a formula to apply in quadratics questions miss its power as a problem-solving tool that earns marks in coordinate geometry, functions, and trigonometry.

Where the discriminant appears in 9709 Paper 1

Topic	Exam Phrasing	Discriminant Condition
Coordinate Geometry	"Show the line is a tangent to the curve"	b² − 4ac = 0
Functions	"Find the range of k for which f(x) = k has no solutions"	b² − 4ac < 0
Quadratics	"Find the values of k for which the equation has two distinct real roots"	b² − 4ac > 0
Trigonometry	Hidden quadratic after substitution — check for valid roots	b² − 4ac ≥ 0 and domain check

The Sketch Method for Quadratic Inequalities

The reliable method for quadratic inequalities: (1) Solve the corresponding equation to find the critical values (roots). (2) Sketch the parabola — is the coefficient of x² positive (opens upward) or negative (opens downward)? (3) Mark the roots on the x-axis. (4) Read the inequality from the graph — shade above the x-axis for 'greater than zero,' shade below for 'less than zero.' Examiners give full credit for a clearly labelled sketch.

Revision Strategy for Quadratics

Priority order: (1) Completing the square — used in quadratics, functions, and coordinate geometry. Start here because it appears the most frequently and errors cascade into other topics. (2) Discriminant conditions — used in coordinate geometry, functions, and trigonometry. Move here once completing the square is reliable. (3) Hidden quadratics — used in trigonometry and algebraic questions. (4) Quadratic inequalities — tested as a standalone skill. You can drill hidden quadratics and inequalities in parallel. Stage 1: learn the methods. Stage 2: practise set-piece questions, five to eight per sub-topic. Stage 3: timed past papers under exam conditions.

A Level Maths Quadratics: Cambridge 9709 Pure 1 Guide