Study Tips
How to Solve Quadratic Equations Three Ways
By Sana Iqbal · · 4 min read
Quick answer
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.
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.
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