GCSE Maths: Harder Quadratics (Higher)

The final questions on a Higher-tier GCSE Mathematics paper very often lead to a quadratic that does not start with a plain $x^2$. This material takes the quadratic work in the category a step further, following the Higher-tier algebra content of the GCSE subject content for England: factorising quadratics of the form $a{x}^2+bx+c$, solving quadratic equations by factorising, by completing the square and with the quadratic formula, equations that must be rearranged into a quadratic first, and the nth term of quadratic sequences.

The quiz has thirteen questions. You factorise quadratics with a leading coefficient greater than 1, including a difference of two squares such as $4x^2 - 25$, solve an equation by factorising, and solve another with the quadratic formula to two decimal places. You use the discriminant to decide how many real roots an equation has, write $2x^2 + 12x + 5$ in completed-square form and read off the turning point, clear a fraction to turn $x+\frac{6}{x}=5$ into a quadratic, solve a rectangle problem and reject the solution that gives a negative length, and find the nth term of a quadratic sequence. The wrong options are built from sign errors in the brackets, completing the square without taking out the factor $a$, and keeping a solution that makes no sense in the context.

The flashcards cover the splitting-the-middle-term method, the quadratic formula, the three cases of the discriminant, completing the square when $a\mathrm{\ne }1$, reading the turning point from completed-square form, and the method for the nth term of a quadratic sequence.

The printable written work has eight problems with full working: two factorisations, one with a common factor to take out first, a solution by factorising, a solution with the formula to two decimal places, completing the square to find a turning point and exact roots, using the discriminant to find an unknown coefficient, an equation with algebraic fractions, a projectile-height problem where both solutions make sense, and a quadratic sequence where you test whether a number is a term.

All of this material is Higher-tier content, and it assumes you are already confident with factorising $x^2+bx+c$ and solving simple quadratics, which the category's linear and quadratic equations material covers. It suits Year 11 students aiming at the top grades.

• Factorise quadratic expressions of the form $ax^2 + bx + c$, including differences of two squares
• Solve quadratic equations by factorising and with the quadratic formula to a given accuracy
• Complete the square when $a \neq 1$ and find turning points
• Use the discriminant to decide the number of real roots
• Rearrange equations into quadratic form and interpret solutions in context
• Find the nth term of a quadratic sequence

Practice material written by Zestly, based on the DfE GCSE mathematics subject content (DFE-00233-2013), Algebra A4, A11, A18 and A25 (Higher-tier content).