B: Algebra and functions: Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.

The idea

This topic covers quadratic functions from multiple angles. You must sketch and interpret their parabolic graphs, identifying the vertex, axis of symmetry, intercepts and whether the curve opens upwards or downwards. Completing the square rewrites ax²+bx+c as a(x+p)²+q, revealing the vertex and enabling solution. The discriminant b²−4ac determines the number of real roots: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots. You solve quadratic equations by factorising, completing the square or using the formula. You also solve equations that are quadratic in a function of the unknown, such as x⁴−5x²+4=0, by substituting u=x².

For example: 2x²−8x+6=0 completes to 2(x−2)²−2, has discriminant 16, and roots x=1 and x=3.

What earns the marks

• Sketch or describe a quadratic graph, showing correct shape, y-intercept and any x-intercepts or vertex.
• Complete the square for a quadratic expression, including when the coefficient of x² is not 1, and use the completed-square form to identify the vertex or solve the equation.
• Calculate the discriminant b²−4ac and interpret its sign to state the number and nature of real roots.
• Solve quadratic equations by an appropriate method, including factorising, completing the square or the quadratic formula, and give exact or decimal answers as required.
• Recognise and solve an equation that is quadratic in a function of the unknown, such as a polynomial in x² or a trigonometric equation, by using a substitution and then solving the resulting quadratic.
• Link the discriminant to graph sketches, for example showing a curve that does not meet the x-axis when the discriminant is negative.

Common mistakes and how to put them right

• Forgetting to divide by a when completing the square if the coefficient of x² is not 1.

  Put it right: Factor out the coefficient of x² from the first two terms before completing the square.

• Misinterpreting the discriminant when it is negative as meaning there are no solutions at all.

  Put it right: A negative discriminant means no real roots, but there may be complex roots; for A-Level Mathematics, state that there are no real roots.

• Solving an equation like x⁴−5x²+4=0 by treating it as a quadratic in x without substitution, leading to errors.

  Put it right: Substitute u=x², solve the quadratic in u, then solve u=x² for x, remembering both positive and negative roots.

• Incorrectly identifying the vertex from completed-square form, for example reading (x−2)²−3 as vertex (2, 3) instead of (2, −3).

  Put it right: The completed-square form a(x−h)²+k has vertex (h, k).

Exam tips

• When asked to sketch a quadratic graph, mark the coordinates of the vertex and any intercepts clearly; a rough sketch with key points is usually sufficient.
• If a question asks for the number of real roots, calculate the discriminant and state its value and sign before concluding; this shows your reasoning.
• For equations quadratic in a function of the unknown, always define your substitution clearly and check that your final answers satisfy the original equation.
• Use completing the square to find the vertex or solve equations when factorising is not obvious; it is a reliable method that also gives exact answers.