What you need to know
Quadratic questions connect algebra and graphs. A strong method is to move confidently between expanded, factorised and completed-square forms, because each form reveals different information about the same parabola.
- Recognise the three useful forms of a quadratic expression.
- Use the discriminant to predict the number of real roots.
- Read the turning point and axis of symmetry from completed-square form.
- Connect roots with x-intercepts and the constant term with the y-intercept.
Key mathematics
General form
y=ax^2+bx+c
Use this form for coefficients, the y-intercept and the discriminant.Completed-square form
y=a(x-h)^2+k
The turning point is (h,k).Discriminant
\Delta=b^2-4ac
Its sign determines how many real roots exist.Worked example
For $y=x^2-6x+5$, complete the square: $$y=(x-3)^2-4$$. The turning point is $(3,-4)$ and the axis of symmetry is $x=3$. Factorising gives $y=(x-1)(x-5)$, so the roots are $x=1$ and $x=5$.
EXAM TECHNIQUE
Exam tips
- Choose the form that matches what the question asks for instead of expanding automatically.
- Sketch key points before drawing a quadratic graph: roots, turning point and y-intercept.
- If a question asks for the number of roots, the discriminant is usually faster than solving the equation.
WATCH OUT
Common mistakes
- Writing the turning point as $(-h,k)$ when the form is $a(x-h)^2+k$.
- Forgetting that the sign of $a$ determines whether the parabola opens upward or downward.
- Using a calculator root without showing the required algebraic method.
Frequently asked questions
What should I know before revising quadratic functions?
You should be comfortable with expanding brackets, factorising simple quadratics and solving linear equations.
Why is completed-square form useful?
It shows the turning point and axis of symmetry immediately and is often the fastest form for graph transformations.
What does a negative discriminant mean?
The quadratic has no real roots, so its graph does not cross the x-axis.