Fundamentals of Business Mathematics and Statistics · Quadratic Equations

Solving Quadratic Equations by Factorisation, Completing the Square and Formula

Updated 10 October 2026 · Fact-checked

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Solving it means finding its two roots. You can factorise it, complete the square, or use the formula x = [−b ± √(b² − 4ac)] ÷ 2a. The formula works for every quadratic.

Understand Solving Quadratic Equations

A quadratic equation is an equation where the highest power of the unknown x is 2. Its standard form is ax² + bx + c = 0, where a, b and c are numbers and a is not zero. If a were zero, the x² term would vanish and you would have a simple linear equation.

The values of x that satisfy the equation are called its roots (or solutions). A quadratic has two roots. They may be different, equal, or not real numbers. At Foundation level you mostly meet real roots.

There are three ways to find the roots. Factorisation splits the expression into two brackets. If (x − p)(x − q) = 0, then either x − p = 0 or x − q = 0, so x = p or x = q. This is the fastest method when the numbers are clean.

Completing the square rewrites the equation as (x + k)² = some number, then takes the square root of both sides. The quadratic formula is what you get when you complete the square on the general equation. That is why it works for every quadratic, even when factorisation is hard.

The key idea behind factorisation is the zero product rule: if two numbers multiply to give zero, at least one of them is zero. This only works when the right side is exactly zero. So always move every term to one side first.

Key formulas to remember

Standard form: ax² + bx + c = 0, a ≠ 0; Bring all terms to the left side so the right side is 0 before you solve.
Zero product rule: If (x − p)(x − q) = 0, then x = p or x = q; Works only when the product equals zero.
Factorisation by splitting the middle term: Find two numbers m and n with m + n = b and m × n = a × c; Split bx into mx + nx, then group terms and take out common factors.
Quadratic formula: x = [−b ± √(b² − 4ac)] ÷ 2a; Works for every quadratic. Put brackets round negative values of b.
Completing the square: x² + bx = (x + b/2)² − (b/2)²; Divide the equation by a first so the coefficient of x² is 1.
Check of roots: Sum of roots = −b ÷ a; product of roots = c ÷ a; Use this to verify your answers quickly.

How to solve Solving Quadratic Equations questions

Use this method for any question that asks you to find the roots of a quadratic equation.

1Bring all terms to one side so the equation reads ax² + bx + c = 0. Clear fractions or brackets first if there are any.
2Identify a, b and c with their signs. If a is negative, you may multiply the whole equation by −1 to make it positive.
3Check whether b² − 4ac is a perfect square, or whether two numbers fit m + n = b and m × n = ac. If yes, factorise.
4Split the middle term, group the terms, take out common factors and write two brackets.
5Put each bracket equal to zero and solve the two simple equations to get both roots.
6If factorising is not clean, use the formula. Substitute a, b and c carefully, find the square root, and work out both signs of ±.
7Verify using sum of roots = −b ÷ a and product of roots = c ÷ a, or by substituting one root back.

Quickest way: Check the options or use sum and product

When to use it: Use this in the exam when the question is an MCQ and the four options list possible roots.

Write the equation in the form ax² + bx + c = 0 and find −b ÷ a and c ÷ a.
Scan the options for a pair of roots whose sum equals −b ÷ a.
Among the remaining pairs, check that the product equals c ÷ a.
If two options still survive, substitute one root into the equation to decide.
For equations that factorise easily, split the middle term directly and read off the roots in under a minute.

Common mistakes in Solving Quadratic Equations

Solving before moving all terms to one side, for example writing x(x − 5) = 6 as x = 6 or x − 5 = 6.
Students apply the zero product rule when the right side is not zero.
Fix: Expand first: x² − 5x − 6 = 0. Then factorise to (x − 6)(x + 1) = 0, so x = 6 or −1.
Giving the wrong signs to the roots, for example writing x = 2 from the bracket (x + 2).
Students copy the number inside the bracket instead of solving the bracket equal to zero.
Fix: Always write x + 2 = 0 and then x = −2.
Using the formula with the wrong sign of b, such as writing −b as −3 when b = −3.
Students rush and do not use brackets for negative numbers.
Fix: Write b in brackets. If b = −3, then −b = −(−3) = 3, and b² = (−3)² = 9.
Taking the square root only with a plus sign and giving one root.
Students forget that both +√ and −√ satisfy the equation.
Fix: Always write ± and calculate two values.
Dividing both sides by x to simplify, such as x² = 5x becoming x = 5.
Cancelling looks like a shortcut.
Fix: Move terms across: x² − 5x = 0, so x(x − 5) = 0 and x = 0 or 5. Dividing by x loses the root x = 0.
Forgetting to divide by a before completing the square.
Students use the x² + bx pattern when the coefficient of x² is not 1.
Fix: Divide every term by a first, then add (half of the new coefficient of x)² to both sides.

Worked examples

Example 1

Solve 2x² − 7x + 3 = 0 by factorisation.

Show the solution

Here a = 2, b = −7, c = 3. Then a × c = 6.
Find two numbers with sum −7 and product 6. They are −6 and −1.
Split the middle term: 2x² − 6x − x + 3 = 0.
Group: 2x(x − 3) − 1(x − 3) = 0.
Take the common bracket: (x − 3)(2x − 1) = 0.
So x − 3 = 0 gives x = 3, and 2x − 1 = 0 gives x = 1/2.
Check: sum = 3 + 0.5 = 3.5 = 7/2 = −b ÷ a. Product = 1.5 = 3/2 = c ÷ a.

Answer: x = 3 or x = 1/2

Example 2

Solve x² − 4x − 1 = 0 using the quadratic formula. Give the answer in surd form.

Show the solution

Here a = 1, b = −4, c = −1.
Discriminant: b² − 4ac = (−4)² − 4(1)(−1) = 16 + 4 = 20.
√20 = 2√5.
x = [−(−4) ± √20] ÷ 2 = (4 ± 2√5) ÷ 2.
Divide each term by 2: x = 2 ± √5.
Check: sum of roots = (2 + √5) + (2 − √5) = 4 = −b ÷ a. Product = 4 − 5 = −1 = c ÷ a.

Answer: x = 2 + √5 or x = 2 − √5

Exam tips

If the equation factorises, do it by factorisation. It is faster than the formula and has fewer sign errors.
Before solving, check whether the equation is given in disguise, such as a fraction equation or a bracket product. Clear it to standard form first.
In MCQs, use sum and product of roots to eliminate options before doing full working.
Watch for options that swap the signs of the roots. These are placed there to catch sign errors.
Remember there is no negative marking, so always mark an answer even if you have to guess between two options.

Practice questions from Quadratic Equations

Solving Quadratic Equations in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationQuadratic Equations and Their Roots

Solving Quadratic Equations: frequently asked questions

Which method is best for solving a quadratic equation?

Use factorisation when two numbers fit easily, because it is quickest. Use the quadratic formula when the numbers are awkward or the roots are surds. The formula works for every quadratic.

What is the difference between factorisation and the quadratic formula method?

Factorisation splits the expression into two brackets and needs suitable whole numbers or simple fractions. The formula needs only a, b and c and always gives the roots. Both give the same answers.

Why do we complete the square if we have the formula?

Completing the square shows where the formula comes from. It is also useful when a question asks you to rewrite the expression in the form (x + k)² plus a constant. In MCQs, the formula or factorisation is usually quicker.

Can a quadratic equation have only one root?

A quadratic has two roots, but they can be equal. For example, x² − 6x + 9 = 0 gives (x − 3)² = 0, so both roots are 3. This is called a repeated root.