A-Level · cie-9709 · Paper 1 (Pure Mathematics 1) · Quadratics · 18 min read · Updated 2026-05-06

Quadratics — A-Level Mathematics Pure 1 Study Guide

For: A-Level Mathematics candidates sitting Paper 1 (Pure Mathematics 1).

Covers: Discriminant, completing the square, solving quadratic equations, quadratic inequalities, quadratic graphs, disguised quadratics, max/min values, and line-quadratic intersection rules.

You should already know: Basic algebra, sketching curves like $y = x^{2}$ , solving linear equations.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the A-Level Mathematics style for educational use. They are not reproductions of past Cambridge International examination papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official Cambridge mark schemes for grading conventions.

1. What Is Quadratics?

A quadratic is a degree-2 polynomial, meaning the highest power of the independent variable (usually $x$ ) is 2. The standard form of a quadratic function is $y = a x^{2} + b x + c$ , where $a \neq = 0$ (if $a = 0$ , the expression reduces to a linear function). Quadratics are also called parabolic functions, as their graphs form a U or inverted U shape called a parabola. This is one of the most frequently tested topics in A-Level Mathematics Paper 1, appearing in both short 2-3 mark questions and longer 5-6 mark problem-solving questions.

2. Discriminant ( $b^{2} - 4 a c$ ) — root nature and tangency

The discriminant of a quadratic $a x^{2} + b x + c = 0$ is derived from the square root term in the quadratic formula, and tells you the number and type of real roots (solutions) the equation has:

If $Δ = b^{2} - 4 a c > 0$ : 2 distinct real roots
If $Δ = b^{2} - 4 a c = 0$ : 1 repeated real root (the quadratic touches the x-axis at exactly one point)
If $Δ = b^{2} - 4 a c < 0$ : No real roots (the parabola never crosses the x-axis)

A special case of the repeated root rule is tangency: if a line intersects a quadratic curve and the resulting quadratic equation has $Δ = 0$ , the line is a tangent to the curve.

Worked Example: Find the values of $p$ for which $p x^{2} + 4 x + 2 = 0$ has no real roots. $Δ = 4^{2} - 4 (p) (2) = 16 - 8 p < 0$ $16 < 8 p ⟹ p > 2$ Exam tip: Examiners often test this rule with unknown constants, and you will lose 1 mark if you forget to state $p \neq = 0$ (though in this case $p > 2$ already satisfies this condition).

3. Completing the square — vertex form $y = a (x - h)^{2} + k$

Completing the square rewrites a quadratic from standard form to vertex form, which lets you directly read off the maximum or minimum point of the parabola. The steps for completing the square are:

Factor out the leading coefficient $a$ from the $x^{2}$ and $x$ terms
Take half the coefficient of the remaining $x$ term, square it, and add and subtract this value inside the bracket
Rewrite the perfect square trinomial as a squared binomial, and simplify the constant terms

Worked Example: Complete the square for $3 x^{2} - 12 x + 7$

Factor out 3 from the first two terms: $3 (x^{2} - 4 x) + 7$
Half of -4 is -2, squared is 4: $3 [(x^{2} - 4 x + 4) - 4] + 7$
Simplify: $3 (x - 2)^{2} - 12 + 7 = 3 (x - 2)^{2} - 5$

Exam tip: Always expand your final vertex form back to standard form to check for arithmetic errors — this takes 10 seconds and avoids losing easy marks.

4. Solving quadratic equations (factoring, formula, completing the square)

There are 3 standard methods to solve quadratic equations, and you will be expected to choose the most efficient method for each question:

Factoring: Use for quadratics with integer roots. For example, $x^{2} - 7 x + 12 = 0$ factors to $(x - 3) (x - 4) = 0$ , so roots are $x = 3$ and $x = 4$ .
Quadratic formula: Use for non-factorable quadratics, especially if the question asks for decimal answers. The formula is: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ For $2 x^{2} - 3 x - 1 = 0$ , roots are $\frac{3 \pm 9 + 8}{4} = \frac{3 \pm 17}{4}$ .
Completing the square: Use if the question asks for exact surd form or the vertex of the parabola. From the earlier example $3 (x - 2)^{2} - 5 = 0$ , roots are $2 \pm \frac{5}{3}$ .

Exam tip: If the question specifies "exact values", do not write decimal approximations — leave answers in surd form for full marks.

5. Quadratic inequalities — sketch + sign analysis

To solve a quadratic inequality, follow these steps:

Rearrange the inequality so all terms are on one side and the right-hand side is 0
Find the roots of the corresponding quadratic equation
Sketch the parabola, remembering that $a > 0$ gives an upward-opening curve, $a < 0$ gives a downward-opening curve
Use the sketch to identify the region(s) that satisfy the inequality: for upward-opening curves, values below the x-axis are between the roots, values above are outside the roots.

Worked Example: Solve $2 x^{2} + x - 6 \geq 0$

Find roots: $2 x^{2} + x - 6 = 0$ factors to $(2 x - 3) (x + 2) = 0$ , so roots $x = - 2$ and $x = \frac{3}{2}$
$a = 2 > 0$ , so the parabola opens upward
The inequality $\geq 0$ corresponds to regions above the x-axis, so the solution is $x \leq - 2$ or $x \geq \frac{3}{2}$ .

Exam trap: Never write combined inequalities like $- 2 \geq x \geq 1.5$ — this is mathematically invalid, and you will lose all marks for the solution.

6. Graph of a quadratic — vertex, axis of symmetry, intercepts

All quadratic graphs have 4 key features you will be expected to identify and label on sketches:

Vertex: The turning point of the parabola, with coordinates $(h, k)$ from vertex form $y = a (x - h)^{2} + k$
Axis of symmetry: The vertical line passing through the vertex, equation $x = h$
Y-intercept: The point where the curve crosses the y-axis, found by substituting $x = 0$ into the standard form (equal to $c$ )
X-intercepts: The points where the curve crosses the x-axis, equal to the real roots of $a x^{2} + b x + c = 0$

Worked Example: State the key features of $y = - 2 x^{2} + 8 x - 5$ (vertex form $y = - 2 (x - 2)^{2} + 3$ )

Vertex: $(2, 3)$
Axis of symmetry: $x = 2$
Y-intercept: $(0, - 5)$
X-intercepts: $2 \pm \frac{3}{2} \approx 0.775$ and $3.225$
Shape: Downward opening, as $a = - 2 < 0$

Exam tip: Always label all intercepts and the vertex on sketches, even if the question does not explicitly ask for them — examiners deduct marks for unlabeled curves.

7. Disguised quadratics — substitutions like $u = x^{2}$ or $u = e^{x}$

Disguised quadratics are equations that are not quadratic in $x$ , but can be rewritten as quadratics using a substitution of the form $u = f (x)$ . Common substitutions include $u = x^{2}$ , $u = \frac{1}{x}$ , and $u = e^{x}$ (note that $u = e^{x}$ is always positive, so negative solutions for $u$ must be discarded).

Worked Example: Solve $x^{4} - 10 x^{2} + 9 = 0$

Substitute $u = x^{2}$ , so the equation becomes $u^{2} - 10 u + 9 = 0$
Factor: $(u - 1) (u - 9) = 0$ , so $u = 1$ or $u = 9$
Substitute back to $x$ : $x^{2} = 1 ⟹ x = \pm 1$ , $x^{2} = 9 ⟹ x = \pm 3$

Exam tip: Never leave answers in terms of the substitution variable $u$ — always reverse the substitution to find values of the original variable, and discard any invalid solutions (e.g. negative $u$ for $u = x^{2}$ ).

8. Maximum and minimum values from vertex form

The vertex form of a quadratic directly gives the maximum or minimum value of the function:

If $a > 0$ (upward opening parabola), the vertex is the minimum point: the minimum value of $y$ is $k$ , occurring at $x = h$
If $a < 0$ (downward opening parabola), the vertex is the maximum point: the maximum value of $y$ is $k$ , occurring at $x = h$

Worked Example: Find the range of the function $f (x) = 3 x^{2} - 12 x + 7$ for all real $x$

Vertex form: $3 (x - 2)^{2} - 5$ , $a = 3 > 0$
Minimum value of $f (x)$ is $- 5$ , no upper limit
Range: $f (x) \geq - 5$

Exam tip: Questions asking for the range of a quadratic function almost always require you to use the vertex form, rather than substituting random values of $x$ .

9. Intersection of a line and a curve — discriminant condition

To find the number of intersection points between a line and a quadratic curve:

Set the equation of the line equal to the equation of the quadratic curve
Rearrange to form a single quadratic equation in $x$ of the form $A x^{2} + B x + C = 0$
Use the discriminant of this new quadratic to find the number of intersections:

$Δ > 0$ : 2 distinct intersection points
$Δ = 0$ : 1 intersection point (the line is tangent to the curve)
$Δ < 0$ : No intersection points

Worked Example: Find the value of $m$ for which the line $y = m x - 2$ is tangent to the curve $y = x^{2} + 3 x + 4$

Set equal: $m x - 2 = x^{2} + 3 x + 4$
Rearrange: $x^{2} + (3 - m) x + 6 = 0$
Tangent so $Δ = 0$ : $(3 - m)^{2} - 4 (1) (6) = 0$ $9 - 6 m + m^{2} - 24 = 0 ⟹ m^{2} - 6 m - 15 = 0$ $m = \frac{6 \pm 36 + 60}{2} = 3 \pm 26$

Exam tip: This is one of the most common 4-5 mark questions in Paper 1 — sign errors when rearranging the equal equations are the leading cause of lost marks here, so double-check your algebra.

10. Common Pitfalls (and how to avoid them)

Wrong move: Forgetting that $a \neq = 0$ when working with quadratics, leading to incorrect solutions for unknown coefficients. Why? Students rush through questions and ignore the definition of a quadratic. Correct move: Always state that the coefficient of $x^{2}$ cannot be zero as a final check, especially if your solution includes a value that makes $a = 0$ .
Wrong move: Combining inequalities for regions outside two roots (e.g. writing $2 \leq x \leq - 3$ instead of $x \leq - 3$ or $x \geq 2$ ). Why? Confusion between regions inside and outside roots. Correct move: For upward-opening parabolas, use AND for regions between roots, OR for regions outside roots.
Wrong move: Leaving solutions for disguised quadratics in terms of $u$ instead of the original variable. Why? Forgetting to reverse the substitution after solving. Correct move: Add a final step to all disguised quadratic questions to substitute back to $x$ and discard invalid solutions.
Wrong move: Sign errors when completing the square for negative leading coefficients. Why? Incorrectly distributing the negative sign when factoring out $a$ . Correct move: Factor out the negative sign first, then complete the square inside the bracket, and carefully expand the constant terms at the end.
Wrong move: Using approximate decimal values when exact surd form is required. Why? Rushing to compute answers with a calculator. Correct move: Check the question wording for "exact" or "3 significant figures" before writing your final answer.

11. Practice Questions (A-Level Mathematics P1 Style)

Question 1 (3 marks)

Find the values of $k$ for which the quadratic equation $2 x^{2} + 3 k x + k = 0$ has a repeated real root.

Solution: For a repeated root, $Δ = 0$ : $Δ = (3 k)^{2} - 4 (2) (k) = 9 k^{2} - 8 k = 0$ $k (9 k - 8) = 0$ Solutions: $k = 0$ or $k = \frac{8}{9}$

Question 2 (5 marks)

a) Complete the square for $y = - x^{2} + 6 x - 4$ (3 marks) b) Hence state the maximum value of $y$ and the $x$ -coordinate at which it occurs. (2 marks)

Solution: a) Factor out -1 from $x$ terms: $- (x^{2} - 6 x) - 4$ Half of -6 is -3, squared is 9: $- [(x^{2} - 6 x + 9) - 9] - 4$ Simplify: $- (x - 3)^{2} + 9 - 4 = - (x - 3)^{2} + 5$ b) $a = - 1 < 0$ , so maximum value of $y$ is 5, occurring at $x = 3$

Question 3 (6 marks)

Find the values of $c$ for which the line $y = 3 x + c$ does not intersect the curve $y = 2 x^{2} - 5 x + 1$ .

Solution: Set equations equal: $3 x + c = 2 x^{2} - 5 x + 1$ Rearrange to standard quadratic form: $2 x^{2} - 8 x + (1 - c) = 0$ No intersection means $Δ < 0$ : $Δ = (- 8)^{2} - 4 (2) (1 - c) = 64 - 8 (1 - c) = 64 - 8 + 8 c = 56 + 8 c < 0$ $8 c < - 56 ⟹ c < - 7$

12. Quick Reference Cheatsheet

Concept	Formula/Rule
Standard Quadratic Form	$a x^{2} + b x + c$ , $a \neq = 0$
Discriminant	$Δ = b^{2} - 4 a c$ : $Δ > 0$ (2 distinct roots), $Δ = 0$ (1 repeated root), $Δ < 0$ (no real roots)
Vertex Form	$y = a (x - h)^{2} + k$ , vertex $(h, k)$ , axis of symmetry $x = h$
Quadratic Formula	$x = \frac{- b \pm Δ}{2 a}$
Quadratic Inequalities (a>0)	$a x^{2} + b x + c < 0$ : solution between roots; $> 0$ : solution outside roots
Max/Min Values	$a > 0$ : min $y = k$ at $x = h$ ; $a < 0$ : max $y = k$ at $x = h$
Line-Quadratic Intersection	$Δ > 0$ (2 intersections), $Δ = 0$ (tangent), $Δ < 0$ (no intersections)

13. What's Next

Quadratics are a foundational topic for the entire A-Level Mathematics syllabus, and you will use the skills from this guide in almost every other Pure 1 topic, including coordinate geometry (circles, parabolas), calculus (finding stationary points and solving derivative equations), and even trigonometric identities. You will also encounter quadratic manipulation in Mechanics 1 kinematics problems and Pure 3 exponential/logarithmic equations, so mastering this topic now will save you significant time later when working on more complex problems.

If you have any questions about specific steps for completing the square, discriminant calculations, or solving exam-style quadratic problems, you can ask Ollie, your AI tutor, at any time for personalized explanations and extra practice. You can also find more A-Level Mathematics Paper 1 study guides and full past paper practice on the homepage.

Aligned with the Cambridge International AS & A Level Mathematics 9709 syllabus. OwlsAi is not affiliated with Cambridge Assessment International Education.

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