Solve quadratic equations showing roots, discriminant, vertex, and step-by-step working
The quadratic formula is one of the most important tools in algebra, providing a reliable method for solving any equation of the form ax^2 + bx + c = 0. Whether you are revising for GCSE maths, studying A-level further maths, or working through university-level problems, understanding how to apply the quadratic formula is essential. A quadratic equation has at most two solutions, called roots. These roots represent the x-values where the corresponding parabola crosses the x-axis. The nature of the roots depends entirely on the discriminant (b^2 - 4ac), which determines whether the equation has two distinct real roots, one repeated root, or two complex conjugate roots. Our calculator solves any quadratic equation instantly, showing the discriminant, both roots (including complex roots where applicable), the vertex of the parabola, and complete step-by-step working. This makes it an ideal revision companion for GCSE and A-level students studying with AQA, Edexcel, OCR, or WJEC exam boards. The quadratic formula appears on the GCSE formulae sheet, but understanding how to use it and interpret the discriminant is essential for higher-tier papers. A-level students need to go further, using the discriminant to determine the nature of roots without solving the full equation, and working with complex roots in further maths modules.
To solve a quadratic equation using this calculator: 1. Enter the coefficient a (the number multiplying x^2). This value must not be zero, otherwise the equation is not quadratic. 2. Enter the coefficient b (the number multiplying x). This can be positive, negative, or zero. 3. Enter the constant c (the standalone number). This can also be positive, negative, or zero. 4. The calculator instantly displays the discriminant, telling you the nature of the roots. A positive discriminant means two real roots. Zero means one repeated root. A negative discriminant means complex roots. 5. Review the roots (x1 and x2) and the vertex coordinates. The vertex shows the turning point of the parabola, useful for graph sketching. 6. Study the step-by-step working to understand how the values are substituted into the quadratic formula. This mirrors the working expected in GCSE and A-level exam answers.
The quadratic formula is: x = (-b +/- sqrt(b^2 - 4ac)) / (2a) The discriminant, D = b^2 - 4ac, determines the nature of the roots: - If D > 0: two distinct real roots, x1 = (-b + sqrt(D)) / (2a) and x2 = (-b - sqrt(D)) / (2a) - If D = 0: one repeated root, x = -b / (2a) - If D < 0: two complex conjugate roots, x = (-b / (2a)) +/- (sqrt(-D) / (2a))i The vertex of the parabola y = ax^2 + bx + c is located at: - x-coordinate: h = -b / (2a) - y-coordinate: k = c - b^2 / (4a) For example, solving x^2 - 5x + 6 = 0: a = 1, b = -5, c = 6. Discriminant = 25 - 24 = 1 (positive, so two real roots). x1 = (5 + 1) / 2 = 3, x2 = (5 - 1) / 2 = 2. The roots are x = 3 and x = 2, which can be verified by factorising as (x - 3)(x - 2) = 0. In GCSE maths (Higher tier), students are expected to solve quadratics by factorising, completing the square, or using the formula. The discriminant is tested explicitly at A-level, where students must determine the number and nature of roots for a given equation.