2.5

Quadratics

Cambridge IGCSE Mathematics (0580)  · Unit 2: Algebra and graphs  · 9 flashcards

Quadratics is topic 2.5 in the Cambridge IGCSE Mathematics (0580) syllabus , positioned in Unit 2 — Algebra and graphs , alongside Algebraic notation and manipulation, Equations and Inequalities.  In one line: The general form is ax² + bx + c = 0, where a, b, and c are coefficients. 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

This topic is examined across Paper 1 (Core) or Paper 2 (Extended) — non-calculator — and Paper 3 (Core) or Paper 4 (Extended) — calculator.

The deck below contains 9 flashcards — 3 definitions, 1 key concept and 1 application card — covering the precise wording mark schemes reward.  Use the 3 definition cards to lock down command-word answers (define, state), then move on to the concept and application cards to handle explain, describe and compare questions.

Key definition

The general form of a quadratic equation, and how do you identify the coefficients

The general form is ax² + bx + c = 0, where a, b, and c are coefficients. 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

Example: in the equation 5x² + 3x - 2 = 0, a = 5, b = 3, and c = -2.

Questions this Quadratics deck will help you answer

Definition Flip

What is the general form of a quadratic equation, and how do you identify the coefficients?

Answer Flip

The general form is ax² + bx + c = 0, where a, b, and c are coefficients. 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

Example: in the equation 5x² + 3x - 2 = 0, a = 5, b = 3, and c = -2.
Key Concept Flip

Explain how to solve a quadratic equation by factorisation. Give an example.

Answer Flip

Factorisation involves expressing the quadratic as a product of two linear factors.

Example: x² + 5x + 6 = (x+2)(x+3) = 0. Solutions are x = -2 and x = -3 (set each factor to zero).
Key Concept Flip

State the quadratic formula and explain when it is most useful.

Answer Flip

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It's most useful when the quadratic equation is difficult or impossible to factorise easily, or to verify factorisation.

Key Concept Flip

What is the discriminant (Δ) of a quadratic equation, and what does it tell you about the roots?

Answer Flip

The discriminant (Δ) is b² - 4ac, calculated from the coefficients of the quadratic equation ax² + bx + c = 0. If Δ > 0, there are two distinct real roots; if Δ = 0, there is one real root (a repeated root); if Δ < 0, there are no real roots.

Example: For x² + 4x + 1 = 0, Δ = 4² - 4(1)(1) = 12 > 0, so there are two distinct real roots.
Definition Flip

Describe the shape of the graph of a quadratic equation (a parabola).

Answer Flip

The graph is a U-shaped curve called a parabola. If the coefficient of x² (a) is positive, the parabola opens upwards, having a minimum point. If 'a' is negative, it opens downwards, having a maximum point.

Example: y = x² opens upwards, while y = -x² opens downwards.
Definition Flip

What is the turning point (vertex) of a parabola, and how does it relate to the minimum or maximum value?

Answer Flip

The turning point, or vertex, is the point where the parabola changes direction. It represents the minimum y-value if the coefficient of x² is positive (opens upwards), and the maximum y-value if it's negative (opens downwards).

Example: the parabola y = x² - 4x + 3 has a turning point at (2, -1), which is the minimum y-value.
Key Concept Flip

Topic: Quadratics. Question: How do you find the axis of symmetry of a parabola given its equation?

Answer Flip

The axis of symmetry is a vertical line passing through the vertex. Its equation is x = -b / 2a, where 'a' and 'b' are coefficients from the quadratic equation in the form ax² + bx + c = 0.

Example: For y = x² + 4x + 3, the axis of symmetry is x = -4 / (2*1) = -2.
Key Concept Flip

Explain the process of completing the square for the quadratic expression x² + 6x + 5. Express in form (x + p)² + q.

Answer Flip

To complete the square: (x + 3)² - 9 + 5 = (x + 3)² - 4. The expression is now in the form (x+p)² + q, where p = 3 and q = -4. This form readily reveals the vertex coordinates.

Key Concept Flip

A quadratic equation has roots at x = 2 and x = -3. Determine the equation in the form ax² + bx + c = 0.

Answer Flip

If roots are 2 and -3, the factors are (x-2) and (x+3). Expanding (x-2)(x+3) gives x² + x - 6 = 0. So, a=1, b=1, and c=-6.

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2.4 Sequences 2.6 Graphs of functions

Key Questions: Quadratics

What is the general form of a quadratic equation, and how do you identify the coefficients?

The general form is ax² + bx + c = 0, where a, b, and c are coefficients. 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

Example: in the equation 5x² + 3x - 2 = 0, a = 5, b = 3, and c = -2.
Describe the shape of the graph of a quadratic equation (a parabola).

The graph is a U-shaped curve called a parabola. If the coefficient of x² (a) is positive, the parabola opens upwards, having a minimum point. If 'a' is negative, it opens downwards, having a maximum point.

Example: y = x² opens upwards, while y = -x² opens downwards.
What is the turning point (vertex) of a parabola, and how does it relate to the minimum or maximum value?

The turning point, or vertex, is the point where the parabola changes direction. It represents the minimum y-value if the coefficient of x² is positive (opens upwards), and the maximum y-value if it's negative (opens downwards).

Example: the parabola y = x² - 4x + 3 has a turning point at (2, -1), which is the minimum y-value.

Tips to avoid common mistakes in Quadratics

More topics in Unit 2 — Algebra and graphs

Quadratics sits alongside these Mathematics decks in the same syllabus unit. Each uses the same spaced-repetition system, so progress in one informs the next.

2.1 Algebraic notation and manipulation

9 flashcards
2.2 Equations

9 flashcards
2.3 Inequalities

9 flashcards
2.4 Sequences

18 flashcards
2.6 Graphs of functions

9 flashcards
2.7 Straight line graphs

9 flashcards
2.8 Functions

9 flashcards
2.9 Differentiation

9 flashcards

Cambridge syllabus keywords to use in your answers

These are the official Cambridge 0580 terms tagged to this section. Mark schemes credit responses that use the exact term — weave them into your answers verbatim rather than paraphrasing.

quadratic x squared factorise quadratic quadratic formula completing the square discriminant roots solutions parabola turning point vertex minimum maximum axis of symmetry

Key terms covered in this Quadratics deck

Every term below is defined in the flashcards above. Use the list as a quick recall test before your exam — if you can't define one of these in your own words, flip back to that card.

The general form of a quadratic equation, and how do you identify the coefficients
Describe the shape of the graph of a quadratic equation (a parabola)
The turning point (vertex) of a parabola, and how does it relate to the minimum or maximum value

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How to study this Quadratics deck

Start in Study Mode, attempt each card before flipping, then rate Hard, Okay or Easy. Cards you rate Hard come back within a day; cards you rate Easy push out to weeks. Your progress is saved in your browser, so come back daily for 5–10 minute reviews until every card reads Mastered.