What are we solving?
A quadratic equation can be written in standard form:
ax² + bx + c = 0 (a ≠ 0)
Its solutions are called roots. A quadratic has 0, 1 or 2 real roots: the x-intercepts of the parabola y = ax² + bx + c.
Method 1: Factoring
Uses the zero product property: if ab = 0, then a = 0 or b = 0.
Example: x² − 5x + 6 = 0
- (x − 2)(x − 3) = 0
- x = 2 or x = 3
Always set the equation equal to zero first.
x² = 4x → x² − 4x = 0 → x(x − 4) = 0 → x = 0 or 4
(Dividing both sides by x would lose the x = 0 root!)
Method 2: Square root method
Works when there’s no x term, or for (x − h)² = k.
Example: 3x² − 48 = 0 → x² = 16 → x = ±4
Example: (x + 1)² = 25 → x + 1 = ±5 → x = 4 or −6
Don’t forget the ±!
Method 3: Quadratic formula
Works for every quadratic:
x = [−b ± √(b² − 4ac)] / (2a)
Example: 2x² + 3x − 2 = 0 (a = 2, b = 3, c = −2)
- b² − 4ac = 9 + 16 = 25
- x = (−3 ± 5)/4 → x = ½ or x = −2
Method 4: Completing the square
Rewrite in the form (x − h)² = k, then use the square root method. (See its own lesson.)
Method 5: Graphing
Graph y = ax² + bx + c and read the x-intercepts. Exact if they’re whole numbers; otherwise approximate.
Choosing a method
| Equation looks like | Best method |
|---|---|
| factors easily | factoring |
| no x term, or (x − h)² = k | square root method |
| doesn’t factor | quadratic formula |
| a = 1 and b is even | completing the square works neatly |
Application
A ball is thrown: h = −4.9t² + 19.6t + 1.5. When does it hit the ground?
−4.9t² + 19.6t + 1.5 = 0 → using the formula, t ≈ 4.07 s (reject the negative root).
Worked examples
Example 1: x² + 2x − 15 = 0 → (x + 5)(x − 3) = 0 → x = −5 or 3.
Example 2: 2x² = 50 → x² = 25 → x = ±5.
Example 3: x² − 4x + 1 = 0 → x = (4 ± √12)/2 = 2 ± √3 (≈ 3.73 or 0.27).
Common mistakes
- Not setting to zero before factoring.
- Forgetting ± with square roots.
- Formula errors: −b, and 2a under the whole numerator.
- Keeping negative times in real problems where they don’t make sense.
Try it yourself
- x² − 9x + 20 = 0
- 4x² − 36 = 0
- x² + 6x = 0
- 3x² − 5x − 2 = 0
- x² + 2x − 5 = 0 (exact)
Answers: 1) 4, 5 2) ±3 3) 0, −6 4) 2, −⅓ 5) −1 ± √6
Quick recap
- Write the quadratic equation in standard form = 0.
- Solve by factoring (zero product property), the square root method or the quadratic formula.
- Check whether roots make sense in context.