"How to solve quadratic equations step by step" is typed by students in Algebra 1 and 2, GCSE, IGCSE and CBSE Class 10, and by parents who learned one method twenty years ago and are being shown another. This page works all three methods in full on real examples, gives the rule for choosing between them from the equation itself, explains what the discriminant tells you before you start, lists the six mistakes examiners see most, and says where each method sits in each curriculum, because the "right" method on a paper is sometimes the one the question names.
Before any method: the shape
A quadratic equation has an x² term and nothing higher. Get everything on one side so it reads ax² + bx + c = 0, with a not zero. So 2x² = 5x + 3 becomes 2x² - 5x - 3 = 0, and the numbers are a = 2, b = -5, c = -3. Every method starts here, and a third of the mistakes come from skipping it (solving 2x² = 5x + 3 by "dividing by x" loses a solution).
Optionally, check the discriminant b² - 4ac. It tells you how many real solutions to expect and, if it is a perfect square, that factoring will work.
Method 1: factoring (when the numbers are friendly)
Example: x² + 5x + 6 = 0. 1. Find two numbers that multiply to c (6) and add to b (5): 2 and 3. 2. Write the brackets: (x + 2)(x + 3) = 0. 3. A product is zero only when a factor is zero: x + 2 = 0 or x + 3 = 0. 4. So x = -2 or x = -3. 5. Check: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0. ✓
Example with a leading coefficient: 2x² - 5x - 3 = 0. 1. Multiply a and c: 2 × (-3) = -6. Find two numbers that multiply to -6 and add to -5: -6 and +1. 2. Split the middle term: 2x² - 6x + x - 3 = 0. 3. Factor in pairs: 2x(x - 3) + 1(x - 3) = 0. 4. Take out the common bracket: (2x + 1)(x - 3) = 0. 5. So x = -1/2 or x = 3.
Factoring is the fastest method when it works, and the one GCSE Foundation and CBSE Class 10 lean on. It fails, honestly, when the discriminant is not a perfect square; then stop trying and use one of the other two.
Method 2: completing the square (for vertex form and exact answers)
Example: x² + 6x + 2 = 0. 1. Move the constant: x² + 6x = -2. 2. Take half of b (6 → 3), square it (9), add to both sides: x² + 6x + 9 = 7. 3. The left side is now a perfect square: (x + 3)² = 7. 4. Square root both sides, remembering both signs: x + 3 = ±√7. 5. So x = -3 + √7 or x = -3 - √7 (about -0.35 and -5.65).
If a is not 1, divide the whole equation by a first. Completing the square is the method that gives the vertex of the parabola for free: (x + 3)² - 7 has its turning point at (-3, -7), which is why GCSE Higher and Algebra 2 ask for it by name. It is also where the quadratic formula comes from, if you complete the square on ax² + bx + c = 0 in general.
Method 3: the quadratic formula (always works)
x = [-b ± √(b² - 4ac)] / 2a
Example: 3x² - 5x - 2 = 0, so a = 3, b = -5, c = -2. 1. Discriminant: b² - 4ac = 25 - 4(3)(-2) = 25 + 24 = 49. Positive and a perfect square, so two real roots (and it would have factored). 2. x = [5 ± √49] / 6 = [5 ± 7] / 6. 3. x = 12/6 = 2 or x = -2/6 = -1/3.
The formula is the safety net: it works for every quadratic, and it is the method the question wants when it says "to 2 decimal places". Two habits keep it accurate: write b and c with their signs before substituting, and work out the discriminant as a separate line.
Which method, from the equation
| The equation looks like | Use | Why |
|---|---|---|
| Small whole numbers, discriminant a perfect square | Factoring | Fastest; two lines of working |
| "Write in the form (x + p)² + q", or "find the vertex / turning point" | Completing the square | The question is asking for that form |
| "Give your answer to 2 decimal places" or in surd form | The formula (or completing the square for surds) | The roots are irrational; factoring cannot find them |
| No x term (x² = 25) | Square root both sides | x = ±5; do not forget the negative |
| No constant (x² + 4x = 0) | Take out x | x(x + 4) = 0, so x = 0 or x = -4; never divide by x |
The six mistakes examiners see most
- Dividing by x and losing the solution x = 0.
- Forgetting the ± when square-rooting.
- Substituting b as positive when it is negative (b = -5 gives -b = +5).
- Halving b but not squaring it (or squaring it but forgetting to add it to both sides) when completing the square.
- Setting one bracket to zero and stopping: a quadratic has two answers unless the discriminant is zero.
- Not writing the equation as ax² + bx + c = 0 first, so a, b and c are read off the wrong form.
Where each method sits in the curriculum
| Course | Methods expected | Notes |
|---|---|---|
| Algebra 1 (US, Grade 8 or 9) | Factoring, square roots, the formula introduced | Vertex form arrives in Algebra 2 |
| Algebra 2 (US, Grade 10 or 11) | All three, plus complex roots when the discriminant is negative | The SAT tests factoring, vertex form and the discriminant, not the formula's arithmetic |
| GCSE Foundation | Factoring with a = 1, the formula | |
| GCSE Higher and IGCSE Extended | All three, completing the square by name, the discriminant | A right answer with no working scores one mark of three |
| CBSE Class 10, chapter 4 | Factoring by splitting the middle term, the formula, the discriminant and the nature of roots | Completing the square is taught but rarely examined; the word problems carry the 5 marks |
Where to get help
A 1-on-1 online maths class with a tutor who holds a master's in mathematics: the three methods on the whiteboard, the choosing rule, then twenty questions of the type the student misses, marked to the real mark scheme. The recording and notes after every class, US$10 to $13 an hour. The Algebra 2, GCSE maths and CBSE Class 9 and 10 maths pages have each course's topics, and the free 30-minute first class is a real lesson on whichever method is going wrong.
Questions parents ask
1What are the steps to solve a quadratic equation?
2How do you solve a quadratic equation by factoring, step by step?
3When should I use the quadratic formula instead of factoring?
4What does the discriminant tell you?
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