Quadratics

Revise Quadratics for Pure Mathematics 1 WMA11 (AS Level) — revision notes and instant AI marking.

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Edexcel IAL · Pure 1

Quadratics

Your complete revision guide — from parabola shapes to hidden quadratics

Big Idea: Every quadratic — no matter how disguised — can be solved by factorising, completing the square, or the quadratic formula. The discriminant tells you how many solutions to expect before you even start.

What's in this chapter?

Quadratic graphs: shape, intercepts, turning point
Discriminant (b²−4ac): counting real roots
Completing the square: rewriting a(x+p)²+q
Solving by factorisation, CTS, quadratic formula
Discriminant inequalities: finding values of parameters
Hidden quadratics: substitution method
Turning point from completed-square form
Proving minimum/maximum values algebraically

1. Quadratic Graphs

Understanding what a quadratic looks like before you do anything else.

The General Form & Parabola Shape

Every quadratic can be written as y = ax² + bx + c, and its graph is always a parabola — a smooth, symmetrical U-shape (or upside-down U). Think of it like a satellite dish: it always has that same curved, symmetric form.

The sign of a (the coefficient of x²) completely determines the orientation:

a > 0 → Positive
U-shape ∪ · Has a minimum point

a < 0 → Negative
∩-shape · Has a maximum point

Quick check: Look at the x² coefficient first — positive = smiling U, negative = frowning ∩. Do this automatically every time you see a quadratic.

Key Features of a Quadratic Graph

You need to be able to find and label four things on a sketch:

y-intercept: Set x = 0. You immediately get y = c. Done.
x-intercepts (roots): Set y = 0. Solve ax² + bx + c = 0 by factorising (or other methods). These are where the curve crosses the x-axis.
Turning point: Complete the square to write as a(x+p)² + q. The turning point is (−p, q).
Line of symmetry: The vertical line through the turning point: x = −p, equivalently x = −b/(2a).

Rearrange first! If the equation isn't in y = ax² + bx + c form, rearrange it before doing anything. Forgetting this loses easy marks.

Worked Example

Sketch y = 2x² + 5x − 12, stating the axes intercepts and turning point.

a = 2 > 0, so it's a positive (U-shaped) parabola.

y-intercept: set x = 0 → y = −12. Point: (0, −12).

x-intercepts: factorise → (2x−3)(x+4) = 0 → x = 3/2 or x = −4. Points: (3/2, 0) and (−4, 0).

Turning point via CTS: y = 2(x + 5/4)² − 121/8. So turning point is (−5/4, −121/8).

Sketch: U-shape, crossing x-axis at −4 and 1.5, y-intercept at −12, minimum below the x-axis.

Practice Question

Sketch the graph of y = −x² + 2x + 8. State the axes intercepts, turning point, and whether it has a max or min.

2. Discriminants

The discriminant is your crystal ball — it tells you how many real solutions exist before you solve anything.

What is the Discriminant?

The discriminant is the expression inside the square root in the quadratic formula. It's given its own symbol, Δ (Greek capital delta), because it carries so much information on its own.

Δ = b² − 4ac

The sign of Δ tells you everything about the roots:

Δ > 0

Δ = 0

Δ < 0

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Also in the full note

What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
Discriminant and Inequalities
What It Means
Method when a = 1
Method when a ≠ 1
Using CTS to Prove Minimum/Maximum

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Quadratics

Quadratics

What's in this chapter?

The General Form & Parabola Shape

Key Features of a Quadratic Graph

What is the Discriminant?

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