Worked example 1
Use the discriminant
How many real roots has x² − 6x + 10 = 0?
- Identify a = 1, b = −6, c = 10
- Calculate Δ = (-6)² − 4(1)(10)
- Compare the value with 0.
Answer: Δ = −4, so there are no real roots
Back to Higher Mathematics
Higher Mathematics
Quadratics and discriminant
Solve quadratics, use the discriminant and connect algebraic form to graph behaviour.
Before you start
- Factorise simple quadratics.
- Use the quadratic formula when needed.
- Know the shape of y = ax² + bx + c
Explanation
Quadratics can be solved by factorising, completing the square or using the quadratic formula. The discriminant b² − 4ac tells you how many real roots the graph has before solving.
Completing the square also reveals the turning point and helps with sketching.
Method and rules
- Discriminant Δ = b² − 4ac
- Δ > 0: two distinct real roots
- Δ = 0: one repeated real root
- Δ < 0: no real roots
- a(x + p)² + q has turning point (-p, q).
Worked examples
Worked example 2
Complete the square
Write x² − 4x + 7 in completed-square form
- Halve −4 to get −2.
- Write (x − 2)²
- Adjust because (x − 2)² = x² − 4x + 4
So: x² − 4x + 7 = (x − 2)² + 3
Watch out
- Using b instead of b² in the discriminant
- Forgetting brackets when b is negative.
- Reading the completed-square turning point with the wrong x sign.
- Using the quadratic formula when factorising would be cleaner.
Exam reminder
If the question asks for nature of roots, the discriminant is enough. If it asks for the roots, solve fully and give exact values when possible.