Quadratic Equations

Understanding quadratic equations

Quadratic equations are an important algebra topic in British secondary school mathematics. Students usually meet simple quadratics at GCSE and IGCSE, then use them more deeply in A-level mathematics. A quadratic equation is an equation where the highest power of the unknown is squared. A typical example is x squared plus 5x plus 6 equals 0. The squared term gives the graph a curved shape called a parabola, and this is why quadratic work connects algebra with graphs. A quadratic equation can often have two solutions, one solution or no real solutions, depending on how the graph meets the x-axis. At GCSE and IGCSE, students usually solve quadratics by factorising, using the quadratic formula, completing the square or reading solutions from a graph. The best method depends on the structure of the question. Some questions are designed to factorise neatly, while others need the formula because the answers are decimals or surds. This topic matters because it develops algebraic fluency and problem-solving confidence. Quadratics appear in questions about area, number patterns, projectile motion, optimisation, graphs and functions. A student who understands quadratic equations properly is better prepared for higher GCSE, IGCSE, A-level algebra and future study in science, economics, computing and engineering.

Key formulas and methods

Standard form: ax squared plus bx plus c equals 0, where a, b and c are numbers and a is not zero. Factorising method: rewrite the quadratic as two brackets, then set each bracket equal to zero. For example, x squared plus 5x plus 6 becomes (x + 2)(x + 3). Therefore x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3. Quadratic formula: x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. This formula works for any quadratic in standard form and is especially useful when factorising is difficult. Discriminant: b squared minus 4ac. If the discriminant is positive, there are two real solutions. If it is zero, there is one repeated real solution. If it is negative, there are no real solutions at GCSE real-number level. Completing the square: this rewrites a quadratic to show its turning point and can help with graph work. For example, x squared plus 6x plus 5 becomes (x + 3) squared minus 4.

Worked examples

Example 1: Solve x squared plus 7x plus 10 equals 0. Find two numbers that multiply to 10 and add to 7. The numbers are 5 and 2. So the equation becomes (x + 5)(x + 2) = 0. Therefore x = -5 or x = -2. Example 2: Solve x squared minus 9 equals 0. This is a difference of two squares. It factorises as (x - 3)(x + 3) = 0. Therefore x = 3 or x = -3. Example 3: Solve 2x squared plus 3x minus 2 equals 0. Factorise as (2x - 1)(x + 2) = 0. Therefore 2x - 1 = 0 or x + 2 = 0. The solutions are x = 1/2 and x = -2.

Practice exercises and answers

Exercise 1: Solve x squared plus 9x plus 20 equals 0. Answer 1: the factorised form is (x + 4)(x + 5) = 0, so x = -4 or x = -5. Exercise 2: Solve x squared minus 16 equals 0. Answer 2: the factorised form is (x - 4)(x + 4) = 0, so x = 4 or x = -4. Exercise 3: Solve 3x squared - 12x = 0. Answer 3: first factorise the common factor. 3x squared - 12x = 3x(x - 4). Therefore 3x(x - 4) = 0, so x = 0 or x = 4.

Where this topic appears in school maths

In GCSE and IGCSE courses, quadratic equations appear in algebra, graphs and problem-solving questions. Students may be asked to solve an equation directly, find where a graph crosses the x-axis or form a quadratic equation from a word problem. For example, a rectangle with algebraic side lengths may lead to a quadratic area equation. Students need to rearrange the equation into standard form before solving. Quadratic graphs are also important. The solutions of a quadratic equation are the x-values where the graph crosses the x-axis. The turning point of the graph can show a minimum or maximum value. This links quadratics to optimisation problems, where students may need to find the smallest area, greatest height or best value. At A-level, quadratics become part of functions, inequalities, transformations, calculus and modelling. Students use the discriminant to reason about roots and may complete the square to find turning points quickly. A strong GCSE understanding makes this later work much easier because the methods are reused in more advanced contexts.

Common mistakes

A common mistake is forgetting that a quadratic can have two solutions. Students sometimes find one bracket value and stop too early. Another mistake is using numbers that multiply correctly but do not add to the middle term. For example, for x squared plus 8x plus 15, the numbers must multiply to 15 and add to 8, so they are 3 and 5. Sign errors are also very common. If the factor is x + 4, the solution is x = -4, not x = 4. Students may also forget to rearrange the equation to equal zero before factorising. If the equation is x squared plus 5x equals 14, it should first become x squared plus 5x minus 14 equals 0. Clear working and checking each step can prevent many lost marks.

Why one-to-one online lessons help

Online maths lessons can be very beneficial for quadratic equations because the topic has several methods, and students need to know which one to choose. A one-to-one tutor can check whether the student understands factorising, standard form, signs, graphs and the quadratic formula. The tutor can then build questions in the correct order, moving from simple factorising to exam-style problem solving. For future study, quadratics are very valuable. They support A-level algebra, functions, calculus, mechanics, physics, economics and data-based modelling. Students who become confident with quadratics often find graph work and advanced algebra less stressful. Regular online tutoring helps students develop accuracy, method choice, exam technique and confidence, so they can move beyond memorising steps and understand why each method works.