Quadratic Equation Solver guide
Factoring, the Formula, or Completing the Square?
Try factoring first — it is fastest when the numbers are small and a = 1. If you cannot spot the factors within about thirty seconds, use the quadratic formula, which always works. Use completing the square when you need vertex form or the question specifically asks for it.
A quick decision rule
- Factoring — when a = 1 and c has obvious factor pairs. Fastest when it works.
- Quadratic formula — when factoring does not come quickly, or the numbers are awkward. Always works.
- Completing the square — when you need the vertex, or the question asks for it by name.
Give factoring thirty seconds
Factoring is genuinely the quickest route when it works. For x² + 5x + 6, spotting that 2 and 3 multiply to 6 and add to 5 takes a few seconds, and you are done.
But it only works cleanly when the discriminant is a perfect square, and you cannot tell that by looking. So set yourself a limit: if the factors have not appeared in about half a minute, stop and use the formula.
Sitting there hunting for factors that do not exist is the most common way to lose time in an exam.
The formula never lets you down
The quadratic formula works on every quadratic — neat ones, ugly ones, ones with irrational roots, ones with complex roots. It never fails and it never requires you to spot anything.
The equation loaded into the calculator, 2x² − 4x − 6 = 0, is a good example of one where the formula is comfortable. With a = 2 the factoring is fiddlier, but the formula handles it exactly as easily as any other.
Its discriminant is 16 + 48 = 64, so the roots are x = 3 and x = −1.
Worth noting: this one could be simplified first. Dividing everything by 2 gives x² − 2x − 3 = 0, which factors easily as (x − 3)(x + 1). Always check for a common factor before deciding a quadratic is hard.
Completing the square is for a different job
As a way of finding roots, completing the square is slower than both other methods, and it is rarely the fastest route to an answer.
Its value is that it produces vertex form, a(x − h)² + k, which hands you the vertex (h, k) directly. If a question asks for the maximum or minimum of a quadratic, this is the method that gives it to you.
It is also where the quadratic formula comes from. Complete the square on the general equation ax² + bx + c = 0 and the formula falls out. Working through that derivation once makes the formula much easier to remember, because you have seen why each piece is there.
Side by side
| Factoring | Formula | Completing the square | |
|---|---|---|---|
| Always works | No | Yes | Yes |
| Speed when it works | Fastest | Medium | Slowest |
| Handles complex roots | No | Yes | Yes |
| Gives the vertex | No | Not directly | Yes |
| Needs you to spot something | Yes | No | No |
What to do if you are stuck
Use the formula. It is the safe default, and there is no penalty for reaching for it.
Students sometimes feel that using the formula on an equation that could have been factored is somehow cheating, or that it will lose marks. It will not. A correct answer is a correct answer.
The only time the method itself is marked is when the question names one — "solve by factoring" or "by completing the square". In that case use what was asked for, then check your answer with the formula if you have time.
