Factoring vs Quadratic Formula vs Completing the Square
Three ways to solve a quadratic equation compared: when factoring is fastest, when to use the formula, and what completing the square shows. Worked examples.
The short answer
All three methods give the same roots for ax² + bx + c = 0; they differ in speed and in what they show. Factoring is fastest when the equation has whole-number or simple fractional roots. The quadratic formula always works, including for irrational and complex roots. Completing the square is the method behind the formula and is the one to use when you need the vertex form of a parabola. A good habit: work out the discriminant b² − 4ac first, then pick the method. The Quadratic Equation Solver shows the discriminant and roots step by step so you can check your work.
Start with the discriminant
The discriminant D = b² − 4ac tells you what kind of roots to expect, and that points to the best method.
| Discriminant | Roots | Best method |
|---|---|---|
| A perfect square (0, 1, 4, 9, 25, ...) | Rational (whole numbers or fractions) | Try factoring first |
| Positive, not a perfect square | Two irrational real roots (with square roots) | Quadratic formula or completing the square |
| Zero | One repeated real root | Factoring (it is a perfect square) |
| Negative | Two complex roots | Quadratic formula |
This assumes a, b and c are integers. For a fuller explanation of the discriminant, see How to Solve a Quadratic Equation Using the Discriminant.
Method 1: Factoring
Rewrite the quadratic as a product of two brackets, then set each bracket to zero.
Example: x² − 5x + 6 = 0
Find two numbers that multiply to 6 and add to −5: −2 and −3.
- (x − 2)(x − 3) = 0
- x = 2 or x = 3
Discriminant check: (−5)² − 4(1)(6) = 25 − 24 = 1, a perfect square, so factoring was a good choice.
Example with a ≠ 1: 3x² − 7x + 2 = 0
Multiply a × c = 6. Find two numbers that multiply to 6 and add to −7: −6 and −1.
- 3x² − 6x − x + 2 = 0
- 3x(x − 2) − 1(x − 2) = 0
- (3x − 1)(x − 2) = 0
- x = 1/3 or x = 2
D = 49 − 24 = 25, a perfect square.
Strengths: quick, no square roots, and you see the factors directly.
Weaknesses: only practical when the roots are rational. If D is not a perfect square, you can search for factors for a long time and never find them.
Method 2: The quadratic formula
x = (−b ± √D) ÷ 2a, where D = b² − 4ac.
Example: x² + 4x − 7 = 0
- a = 1, b = 4, c = −7
- D = 16 − 4(1)(−7) = 16 + 28 = 44
- x = (−4 ± √44) ÷ 2 = −2 ± √11
- x ≈ 1.3166 or x ≈ −5.3166
44 is not a perfect square, so factoring with whole numbers would never have worked.
Example with complex roots: x² + 2x + 5 = 0
- D = 4 − 20 = −16
- x = (−2 ± √−16) ÷ 2 = (−2 ± 4i) ÷ 2
- x = −1 + 2i or x = −1 − 2i
Strengths: works for every quadratic, mechanical and reliable.
Weaknesses: more arithmetic, so sign errors are common, especially with negative b or c. Always write b² with brackets: (−5)² = 25, not −25.
Method 3: Completing the square
Rewrite the equation in the form (x + p)² = q, then take square roots.
Example: x² + 4x − 7 = 0
If a is not 1, divide the whole equation by a first.
Strengths: gives the vertex form. From (x + 2)² − 11 = 0 you can read that the parabola y = x² + 4x − 7 has its lowest point at (−2, −11). It is also how the quadratic formula is derived, so understanding it makes the formula less mysterious.
Weaknesses: slower for routine solving, and fractions appear quickly when b is odd or a ≠ 1.
Side-by-side comparison
| | Factoring | Quadratic formula | Completing the square | |
|---|---|---|---|
| Works for all quadratics | No, only rational roots | Yes | Yes |
| Speed for simple cases | Fastest | Medium | Slowest |
| Handles complex roots | No | Yes | Yes |
| Shows the vertex | No | No | Yes |
| Typical exam use | "Solve by factorisation" | "Give answers to 2 decimal places" | "Express in the form (x + p)² + q" |
Exam wording often tells you which method is expected. If a question asks for answers to a number of decimal places, the roots are probably irrational, which is a strong hint to use the formula.
A simple decision routine
Checking answers with the solver
The solver does not accept a = 0, because then the equation is linear, not quadratic. It gives decimal answers, so if your teacher wants exact forms such as −2 ± √11, keep your working and use the solver only to confirm the decimals.
| Equation | D | Solver result |
|---|---|---|
| x² − 5x + 6 = 0 | 1 | 3 and 2 |
| 3x² − 7x + 2 = 0 | 25 | 2 and 0.3333 |
| x² − 6x + 9 = 0 | 0 | 3 (repeated) |
| x² + 4x − 7 = 0 | 44 | 1.3166 and −5.3166 |
| x² + 2x + 5 = 0 | −16 | −1 ± 2i |
Common mistakes with any method
- Forgetting to move every term to one side before solving.
- Dropping the ± and finding only one root.
- Dividing only part of the equation by a.
- Writing −b as b when b is already negative.
For quick arithmetic with square roots or powers, the Scientific Calculator helps.
Frequently asked questions
When should I use factoring instead of the quadratic formula?
When the discriminant b² − 4ac is a perfect square (and a, b, c are integers), the roots are rational and factoring is usually fastest. Otherwise use the formula.
Does the quadratic formula always work?
Yes. It solves every quadratic with a ≠ 0, including those with irrational or complex roots.
Why learn completing the square?
It gives the vertex form (x + p)² + q, which shows the turning point of the parabola, and it is how the quadratic formula is derived.
What are the roots of x² + 4x − 7 = 0?
x = −2 ± √11, which is about 1.3166 and −5.3166. The discriminant is 44.
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