How to solve quadratic equations using the quadratic formula in 2026
Quadratic equations are polynomial equations of degree two, written in the standard form ax squared plus bx plus c equals zero, where a, b and c are constants and a is not equal to zero. They are one of the most important topics in GCSE and A-level mathematics in the United Kingdom and appear throughout science, engineering and economics. In 2026, every student studying mathematics will encounter quadratic equations and need to know how to solve them.
The most powerful and general method for solving quadratic equations is the quadratic formula. This formula states that the solutions (or roots) of ax squared plus bx plus c equals zero are given by x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. This formula works for every quadratic equation, regardless of whether the roots are integers, fractions, irrational numbers or complex numbers.
To use the quadratic formula, you first identify the coefficients a, b and c from your equation. Make sure the equation is in standard form with all terms on one side and zero on the other. Then substitute the values into the formula and simplify. The plus-or-minus sign means you perform the calculation twice: once with addition to find one root and once with subtraction to find the other.
For example, consider the equation x squared minus 5x plus 6 equals zero. Here a equals 1, b equals negative 5 and c equals 6. Substituting into the formula gives x equals (5 plus or minus the square root of 25 minus 24) divided by 2, which simplifies to x equals (5 plus or minus 1) divided by 2. This gives x equals 3 and x equals 2 as the two roots.
The quadratic formula is derived by a technique called completing the square, which transforms the general quadratic into a form where x can be isolated. Starting from ax squared plus bx plus c equals zero, divide through by a, move the constant to the right side, add (b divided by 2a) squared to both sides, factor the left side as a perfect square and then solve for x. This derivation is often examined at GCSE level and provides insight into why the formula works.
While the quadratic formula is the most reliable method, there are situations where other approaches are more efficient. If the quadratic factorises neatly with integer roots, factorisation is usually quicker. If the equation is already in or close to vertex form, completing the square may be more natural. However, the quadratic formula always works and never fails, making it the go-to method when other approaches prove difficult.
An important consideration when using the quadratic formula is numerical precision. When b squared is very close to 4ac, rounding errors can affect the accuracy of the roots. In computational applications, special techniques such as using the alternative form of the quadratic formula (which avoids catastrophic cancellation) are sometimes necessary. For typical examination and homework problems, however, standard calculator arithmetic provides more than adequate precision.
Understanding the discriminant and the nature of quadratic roots
The discriminant is one of the most useful concepts in the study of quadratic equations. Denoted by the Greek letter delta or simply by the expression b squared minus 4ac, the discriminant tells you everything about the nature of the roots without requiring you to actually solve the equation. This makes it an invaluable tool for analysis, classification and examination questions in the UK mathematics curriculum in 2026.
When the discriminant is positive (greater than zero), the quadratic equation has two distinct real roots. The two solutions are different numbers, and the graph of the corresponding quadratic function (a parabola) crosses the x-axis at two distinct points. The further the discriminant is from zero, the more widely separated the two roots are. For example, the equation x squared minus 4 equals zero has a discriminant of 16, and its roots (x equals 2 and x equals negative 2) are four units apart.
When the discriminant is exactly zero, the quadratic equation has exactly one real root (a repeated or double root). Both solutions given by the quadratic formula are the same value, and the graph of the parabola just touches the x-axis at a single point (the vertex). This situation is sometimes called a perfect square because the quadratic can be written as a times (x minus r) squared. For example, x squared minus 6x plus 9 equals zero has a discriminant of zero and the repeated root x equals 3.
When the discriminant is negative (less than zero), the quadratic equation has no real roots. Instead, it has two complex conjugate roots of the form p plus qi and p minus qi, where i is the imaginary unit defined as the square root of negative one. In this case, the graph of the parabola does not cross the x-axis at all, sitting entirely above (if a is positive) or entirely below (if a is negative) the x-axis. Complex roots are introduced at A-level Further Mathematics and are fundamental in engineering, physics and advanced mathematics.
The discriminant has numerous applications beyond simply classifying roots. In coordinate geometry, it can determine whether a line intersects a curve. By substituting the equation of a line into a quadratic (or a circle equation) and examining the discriminant of the resulting quadratic, you can determine whether the line crosses the curve at two points (discriminant positive), touches it tangentially at one point (discriminant zero) or misses it entirely (discriminant negative).
In optimisation problems, the discriminant helps identify the conditions under which equations have real solutions. For example, if a problem requires real values of a parameter for a physical situation to be possible, setting the discriminant greater than or equal to zero and solving the resulting inequality gives the valid range of the parameter. This technique is commonly examined at A-level and appears in physics, engineering and economics contexts.
The discriminant also connects to the concept of the sum and product of roots through Vieta's formulae. For the equation ax squared plus bx plus c equals zero, the sum of the roots equals negative b over a and the product of the roots equals c over a. Combined with the discriminant, these relationships allow you to determine a great deal about the roots without solving the equation explicitly.
The vertex, axis of symmetry and graph of a quadratic function
Every quadratic equation ax squared plus bx plus c equals zero corresponds to a quadratic function y equals ax squared plus bx plus c whose graph is a parabola. Understanding the key features of this parabola, particularly the vertex, axis of symmetry and direction of opening, is essential for sketching graphs and solving problems in the UK mathematics curriculum in 2026.
The vertex is the turning point of the parabola, the point where the curve changes direction. For a parabola that opens upward (when a is positive), the vertex is the minimum point. For a parabola that opens downward (when a is negative), the vertex is the maximum point. The x-coordinate of the vertex is given by the formula negative b divided by 2a, and the y-coordinate is found by substituting this x-value back into the original equation.
The axis of symmetry is the vertical line that passes through the vertex, with the equation x equals negative b divided by 2a. Every parabola has exactly one axis of symmetry, and the two roots of the equation (if they exist) are equidistant from this line. This symmetry property is useful for sketching: once you have found the vertex and one other point on the parabola, you can reflect it across the axis of symmetry to find its mirror image.
The vertex form of a quadratic function is y equals a times (x minus h) squared plus k, where (h, k) is the vertex. Converting from standard form to vertex form is done by completing the square. For example, y equals x squared minus 6x plus 5 can be rewritten as y equals (x minus 3) squared minus 4, revealing that the vertex is at (3, negative 4). The vertex form makes it easy to identify the vertex, the axis of symmetry (x equals h) and the minimum or maximum value of the function (y equals k).
The y-intercept of the parabola is the point where it crosses the y-axis, found by setting x equals zero. This gives y equals c, so the y-intercept is always (0, c). The x-intercepts (if they exist) are the roots of the equation ax squared plus bx plus c equals zero, found using the quadratic formula or factorisation.
The coefficient a determines how wide or narrow the parabola is. When the absolute value of a is large (greater than 1), the parabola is narrow (steep). When the absolute value of a is small (between 0 and 1), the parabola is wide (shallow). The sign of a determines the direction: positive a opens upward (U-shaped) and negative a opens downward (inverted U-shaped).
Sketching a complete parabola requires identifying several key features: the direction of opening (sign of a), the vertex (using negative b over 2a), the axis of symmetry (x equals negative b over 2a), the y-intercept (0, c), the x-intercepts (if any, using the quadratic formula) and the general shape (narrow or wide, based on the magnitude of a). With these features identified, you can draw an accurate sketch without plotting numerous individual points.
Applications of the vertex and parabola extend far beyond pure mathematics. In physics, projectile motion follows a parabolic path, and the vertex gives the maximum height reached. In engineering, parabolic reflectors (used in satellite dishes, car headlights and solar concentrators) exploit the reflective properties of parabolas. In business and economics, quadratic cost and revenue functions have vertices that represent minimum cost or maximum revenue points, making them essential for optimisation decisions.
Methods of solving quadratic equations and when to use each one
There are several methods for solving quadratic equations, each with its own strengths and ideal use cases. Understanding when to use each method saves time in examinations and builds deeper mathematical fluency. Here is a practical guide to the main techniques taught in UK schools in 2026, along with advice on choosing the most efficient approach.
Factorisation is often the quickest method when it works. To factorise ax squared plus bx plus c, you need to find two numbers that multiply to give ac and add to give b. If such numbers exist as integers, the quadratic can be expressed as a product of two linear factors. For example, x squared minus 7x plus 12 factors as (x minus 3)(x minus 4) because 3 times 4 equals 12 and 3 plus 4 equals 7. Setting each factor equal to zero gives the roots x equals 3 and x equals 4. Factorisation is best used when the coefficients are small integers and the roots are rational numbers. It is the preferred method in GCSE examinations when the question specifically asks you to factorise.
Completing the square is a powerful technique that transforms the quadratic into vertex form. Starting from ax squared plus bx plus c equals zero, you rearrange to isolate a perfect square on one side. For example, x squared plus 6x plus 5 equals zero becomes (x plus 3) squared minus 9 plus 5 equals zero, then (x plus 3) squared equals 4, giving x plus 3 equals plus or minus 2, so x equals negative 1 or x equals negative 5. This method is particularly useful when you need to find the vertex of the parabola, express the quadratic in vertex form or derive results in proof and derivation questions. It is also the method used to prove the quadratic formula itself.
The quadratic formula is the most general and reliable method. It works for every quadratic equation without exception, including those with irrational or complex roots. Its disadvantage is that it involves more computation than factorisation or completing the square for simple cases. The quadratic formula is the best choice when factorisation is not obvious, when the coefficients are large or non-integer, or when you need to handle the general case in algebraic proofs.
Graphical methods involve plotting the quadratic function and reading off the x-intercepts. While this approach gives approximate answers rather than exact ones, it provides valuable visual insight into the behaviour of the equation. Modern graphing calculators and software make this approach quick and practical. In examinations, graphical methods are sometimes used alongside algebraic methods to check answers or to solve problems that are presented graphically.
Trial and improvement (also called trial and error) is rarely used for quadratic equations because the other methods are more efficient. However, for non-standard equations or numerical problems where exact solutions are not required, iterative methods can be effective. The Newton-Raphson method, taught in A-level Further Mathematics, is a powerful iterative technique that can solve not only quadratic equations but equations of any degree.
When deciding which method to use in an examination, consider the following practical guidelines. If the question says factorise, use factorisation. If the question asks for exact solutions and the quadratic does not factorise easily, use the quadratic formula. If the question asks for the vertex or turning point, use completing the square. If the question involves a parameter or general coefficients, the quadratic formula or completing the square are usually the best approaches.
One common examination technique is to use the discriminant as a preliminary check before solving. Calculating b squared minus 4ac takes only a few seconds and immediately tells you whether to expect two real roots, one repeated root or complex roots. This helps you choose the right method and also helps you check your answer, since you already know what type of solution to expect.
Practice is the key to becoming proficient with quadratic equations. Working through a variety of problems, from simple factorisation to complex formula applications, builds the fluency and pattern recognition needed to tackle examination questions efficiently. This calculator is a useful tool for checking your work and building confidence with the different methods.