Quadratic Equations
Learn to factorise quadratics, apply the quadratic formula, and graph parabolas by identifying key features.
Standard Form
A quadratic equation is any equation that can be written in the form:
ax2 + bx + c = 0
a, b, and c are constants (numbers), where a ≠ 0
The highest power of x is 2 — this is what makes it "quadratic"
Key Point: A quadratic equation can have 2 solutions, 1 solution (repeated root), or no real solutions.
Factorising Quadratics
Factorising means rewriting the quadratic as a product of two brackets. We look for two numbers that multiply to give c and add to give b (when a = 1).
Factorising Process (when a = 1)
x2 + bx + c = (x + p)(x + q)
where p + q = b and p × q = c
Step 1
Find two numbers that multiply to c
Step 2
Check which pair also adds to b
Step 3
Write as (x + p)(x + q) = 0
The Quadratic Formula
When a quadratic cannot be easily factorised, use the quadratic formula:
x = −b ± √(b2 − 4ac) 2a
The Discriminant: b2 − 4ac
The expression under the square root determines how many solutions exist:
b2 − 4ac > 0
2 distinct real solutions
b2 − 4ac = 0
1 repeated solution
b2 − 4ac < 0
No real solutions
Graphing Parabolas
The graph of a quadratic equation is called a parabola. Its key features are the vertex, axis of symmetry, and roots (x-intercepts).
Vertex
The turning point (highest or lowest point). For y = ax2 + bx + c, the x-coordinate of the vertex is x = −b / 2a.
Axis of Symmetry
The vertical line through the vertex: x = −b / 2a. The parabola is symmetric about this line.
Roots / X-intercepts
The points where the parabola crosses the x-axis (where y = 0). Found by solving ax2 + bx + c = 0.
Graph of y = x2 − 4x + 3
y = x2 − 4x + 3 has roots at x = 1 and x = 3, vertex at (2, −1), and axis of symmetry x = 2.
Key Vocabulary
| Term | Definition |
|---|---|
| Quadratic | An expression or equation where the highest power of the variable is 2. |
| Factorise | Rewrite an expression as a product of simpler expressions (factors). |
| Parabola | The U-shaped curve produced by graphing a quadratic equation. |
| Vertex | The turning point (minimum or maximum) of a parabola. |
| Discriminant | The value b2 − 4ac that determines the number of real solutions. |
| Roots | The solutions to the equation; the x-values where the parabola crosses the x-axis. |
Worked Examples
Solve by factorising: x2 + 5x + 6 = 0
Step 1: Find two numbers. We need p × q = 6 and p + q = 5. The numbers are 2 and 3.
Step 2: Factorise. x2 + 5x + 6 = (x + 2)(x + 3) = 0
Step 3: Solve each bracket. x + 2 = 0 ⇒ x = −2, or x + 3 = 0 ⇒ x = −3
Solutions: x = −2 or x = −3
Solve using the quadratic formula: 2x2 − 3x − 5 = 0
Step 1: Identify a, b, c. a = 2, b = −3, c = −5
Step 2: Calculate discriminant. b2 − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49
Step 3: Apply formula. x = (3 ± √49) / (2 × 2) = (3 ± 7) / 4
Step 4: Two solutions. x = (3 + 7)/4 = 10/4 = 2.5, or x = (3 − 7)/4 = −4/4 = −1
Find the vertex and axis of symmetry: y = x2 − 6x + 8
Step 1: Find x-coordinate of vertex. x = −b / 2a = −(−6) / (2 × 1) = 6/2 = 3
Step 2: Find y-coordinate. y = (3)2 − 6(3) + 8 = 9 − 18 + 8 = −1
Vertex: (3, −1)
Axis of symmetry: x = 3
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see feedback.
Question 1
Factorise: x2 − 7x + 12 = 0. What are the solutions?
Question 2
What is the discriminant of 3x2 + 2x + 5 = 0?
Question 3
What is the vertex of y = x2 − 2x − 3?
Question 4
If the discriminant equals zero, how many solutions does the quadratic have?
Question 5
Solve x2 − 9 = 0.
Key Concepts Summary
- ● Standard form: ax2 + bx + c = 0 where a ≠ 0.
- ● Factorising: Find two numbers that multiply to c and add to b (when a = 1).
- ● Quadratic formula: x = (−b ± √(b2 − 4ac)) / 2a works for all quadratics.
- ● The discriminant (b2 − 4ac) tells you how many solutions exist.
- ● A parabola's vertex is at x = −b/2a; the axis of symmetry passes through the vertex.