GCSE Maths: Linear and Quadratic Equations

Solving equations is where GCSE algebra earns its marks. This quiz covers the two central families — linear and quadratic — in a progression that mirrors how the skills are assessed across both papers.

The linear questions start with unknowns on both sides (5x − 7 = 2x + 8) and equations with brackets, including the expansion trap 4x − 2(x − 3), where the distributed negative sign catches a large share of candidates. A worded question about a rectangle's perimeter tests the modelling step the specification calls "deriving an equation from a situation": you must recognise which equations correctly represent the constraint before any solving happens.

The quadratic questions then work through the standard ladder: solving by factorising, using the difference of two squares to solve x² − 9 = 0 with both roots, simplifying 2x² + 8x − 10 = 0 by dividing through by the common factor before factorising, and solving equations already in completed-square form like (x + 2)² = 16 — where the key insight is that square-rooting produces both a positive and a negative case. Higher tier techniques appear through completing the square of x² + 6x + 5 and a reverse-engineering question that asks which equation has a given pair of roots, forcing you to rebuild the quadratic from its factors.

Every question is self-contained with concrete numbers, each explanation shows the full method, and the wrong options are constructed from the classic mistakes: sign slips when collecting terms, forgetting the negative root, and factorising with the wrong sign pair. Together with the Expanding and Factorising set, this pair covers the algebra core that grades 4 through 7 are built on.

• Solve linear equations with unknowns on both sides and with brackets
• Form an equation from a worded geometric or real-life situation
• Solve quadratic equations by factorising, including difference-of-two-squares cases
• Simplify quadratics by dividing through by a common factor before factorising
• Use completed-square form, remembering both positive and negative roots

Topic scope follows the Algebra strand of the Department for Education's GCSE mathematics subject content: solving linear equations in one unknown algebraically, solving quadratic equations by factorising and completing the square, and deriving equations from situations before solving them.