Try factorising first — it is fastest when the quadratic factorises neatly. If it does not, use the quadratic formula, which always works. Completing the square is for finding turning points and for proofs. The discriminant tells you how many solutions exist before you solve.
What a quadratic is asking
A quadratic equation sets a curved expression equal to zero, and solving it means finding where the curve crosses the x-axis. A parabola can cross twice, touch once, or miss entirely — which is why a quadratic has two, one, or no real solutions.
Knowing that in advance stops you from hunting for a second solution that does not exist, or stopping at one when there are two.
Factorising first
If you can write the quadratic as two brackets multiplied together, each bracket set to zero gives a solution. This is the fastest method when it works, and it works whenever the solutions are simple.
Practise spotting the two numbers that multiply to give the constant and add to give the middle coefficient. When they exist and are whole, factorising beats every other method for speed.
The quadratic formula always works
When factorising fails — usually because the answers are not whole numbers — the quadratic formula solves any quadratic. Substitute a, b and c carefully, and mind the signs, which is where the errors live.
Because it always works, some students use it for everything. That is fine, but it is slower than factorising when the quadratic factorises neatly, so it is worth trying to factorise first.
Completing the square
Completing the square rewrites the quadratic to reveal the vertex — the turning point of the parabola. It is essential for sketching, for finding maximum and minimum values, and for deriving the quadratic formula itself.
It is less often the quickest way to solve, but questions frequently ask for it by name, so it is worth being fluent in even when the formula would be faster.
Use the discriminant first
The part of the formula under the square root — b squared minus 4ac — tells you how many solutions exist before you do any solving. Positive means two, zero means one, negative means none real.
Checking it first saves time and marks: it tells you what to expect, and 'show there are no real solutions' questions are answered by the discriminant alone.
Watch the setup
Every method assumes the quadratic equals zero. If the equation is not already in that form, rearrange it first — a surprising number of errors come from applying the formula to an equation that was not set to zero.
Also keep a, b and c with their signs. A negative b that becomes positive in the formula is a classic slip.
What the formula does
The quadratic formula gives the solutions of any quadratic equation directly from its coefficients. Unlike factorising, it always works, including for equations that do not factorise neatly.
This universality is its value. When factorising fails or is hard to spot, the formula reliably delivers the solutions.
Identify the coefficients
The first step is identifying the three coefficients from the equation, with their correct signs. A negative coefficient must be carried as negative into the formula, which is where many errors begin.
Writing out the three values clearly, signs included, before substituting prevents a large share of mistakes.
Substitute carefully
Substituting the coefficients into the formula demands care, especially with signs and with squaring. A single sign error propagates through the whole calculation to a wrong answer.
Going slowly at the substitution stage, and double-checking each value, is worth the time. Speed here causes most of the errors.
Handle the part under the root
The expression under the square root, the discriminant, is a common trouble spot — particularly getting its sign right and evaluating it correctly. It also tells you how many solutions there are.
A negative value under the root means no real solutions; zero means one; positive means two. Understanding this adds meaning and provides a useful check.
Work through in steps
Rather than trying to do the whole formula at once, evaluate it in stages — the discriminant first, then the square root, then the two solutions. This structured approach reduces errors.
Breaking the calculation into steps, and checking each, turns an error-prone formula into a reliable procedure.
When to use it
The formula is best when an equation does not factorise easily or at all. For equations that factorise cleanly, factorising is often quicker, so recognising which method suits an equation saves time.
Knowing the formula is your reliable fallback, and using it when factorising is not obvious, is part of handling quadratics efficiently.
Frequently asked questions
Which method should I use?
Try factorising first — it is fastest when the answers are whole numbers. If it does not factorise cleanly, use the quadratic formula, which always works. Use completing the square when asked for it or for turning points.
What does the discriminant tell me?
The discriminant (b² − 4ac) tells you how many real solutions exist: positive means two, zero means one repeated, and negative means none. Checking it first tells you what to expect.
Why is my quadratic formula answer wrong?
Almost always a sign error, or the equation was not set to zero first. Write a, b and c with their signs before substituting, and double-check the −b at the front.
When do I use completing the square?
For finding the turning point of a parabola, for maximum and minimum problems, and whenever a question asks for it by name. It is also how the quadratic formula is derived.
Can a quadratic have no solution?
It can have no real solution — when the discriminant is negative, the parabola does not cross the x-axis. At higher levels these are handled with complex numbers.
What is the quadratic formula for?
Solving any quadratic equation directly from its coefficients, including those that do not factorise neatly. Its value is that it always works, unlike factorising, which needs the equation to factorise.
What's the most common mistake with the formula?
Sign errors and mishandling the part under the square root. A negative coefficient carried in wrongly, or a sign slip, propagates through the whole calculation, so careful, methodical working pays off.
What does the part under the square root tell me?
The discriminant shows how many solutions there are — negative means no real solutions, zero means one, positive means two. Understanding this adds meaning and provides a useful check on your answer.
