Quadratic Equation definition
A quadratic equation is an equation where the highest power of the variable is 2. This means it contains a squared term, such as x squared. A common form is ax squared plus bx plus c equals 0, where a, b and c are numbers and a is not 0. If a were 0, the squared term would disappear and the equation would become linear instead of quadratic.
Examples of quadratic equations include x squared plus 5x plus 6 equals 0, 2x squared minus 3x equals 5, and x squared minus 9 equals 0. The important feature is the x squared term. Quadratic equations can have two solutions, one repeated solution or no real solutions. These solutions are sometimes called roots because they are the x-values that make the equation true.
Quadratic equations are used in algebra, graphs, functions, factorisation, coordinate geometry, modelling, physics, business and optimisation. At GCSE level, students solve quadratics by factorising, using the quadratic formula, completing the square or reading solutions from graphs. At A-Level, quadratics appear in functions, transformations, inequalities, calculus, mechanics and more advanced algebraic modelling.
One common solving method is factorising. For example, x squared plus 5x plus 6 equals 0 factorises to x plus 2 multiplied by x plus 3 equals 0. If two factors multiply to make 0, at least one factor must be 0. Therefore, x plus 2 equals 0 or x plus 3 equals 0. The solutions are x equals -2 and x equals -3.
The graph of a quadratic function is called a parabola. It has a curved U shape or upside-down U shape depending on the sign of the x squared coefficient. Where the graph crosses the x-axis, the y-value is 0, so those crossing points are the solutions of the related quadratic equation. This link between equations and graphs is important for understanding roots visually.
The quadratic formula can solve any quadratic equation in standard form. It uses the values of a, b and c from ax squared plus bx plus c equals 0. The formula is useful when a quadratic does not factorise easily. Students must substitute carefully because sign errors are common. The expression under the square root can also show how many real solutions the equation has.
Quadratic equations can model real situations where change is not constant. They appear in projectile motion, areas of rectangles, profit models, braking distance and optimisation problems. For example, the height of a thrown object over time can often be modelled by a quadratic because gravity changes the motion. This makes quadratics useful beyond pure algebra.
Related ideas include factorising, expanding brackets, roots, parabolas, quadratic formula, completing the square, discriminant, functions and graph intersections. A clear definition of quadratic equation helps students recognise the x squared structure and choose the correct solving method. In exams, first rearrange the equation into standard form, then decide which solving method is most suitable.