The formula
For any quadratic in standard form ax² + bx + c = 0:
x = [−b ± √(b² − 4ac)] / (2a)
It always works, whether or not the quadratic factors.
Steps
- Write the equation as ax² + bx + c = 0.
- Identify the coefficients a, b and c (with their signs).
- Calculate the discriminant b² − 4ac.
- Substitute into the formula.
- Simplify both solutions (+ and −).
Example 1: two rational roots
2x² − 7x + 3 = 0 (a = 2, b = −7, c = 3)
- b² − 4ac = 49 − 24 = 25
- x = (7 ± 5)/4
- x = 3 or x = ½
Example 2: irrational roots
x² + 4x − 1 = 0 (a = 1, b = 4, c = −1)
- b² − 4ac = 16 + 4 = 20
- x = (−4 ± √20)/2 = (−4 ± 2√5)/2 = −2 ± √5
- Decimal: x ≈ 0.236 or x ≈ −4.236
Leave answers in exact form unless asked to round.
Example 3: one root
x² − 6x + 9 = 0
- b² − 4ac = 36 − 36 = 0
- x = 6/2 = 3 (a double root)
Example 4: no real roots
x² + 2x + 5 = 0
- b² − 4ac = 4 − 20 = −16 < 0
- No real roots (the parabola doesn’t touch the x-axis).
What the discriminant tells you
| b² − 4ac | Real roots |
|---|---|
| > 0 | 2 different |
| = 0 | 1 (repeated) |
| < 0 | none |
Rearranging first
3x² = 5x + 2 → 3x² − 5x − 2 = 0 → a = 3, b = −5, c = −2.
Application
A rectangle’s length is 3 m more than its width, and its area is 50 m². Find the width.
- w(w + 3) = 50 → w² + 3w − 50 = 0
- w = (−3 + √209)/2 ≈ 5.73 m (reject the negative)
Worked examples
Example 1: x² − 3x − 10 = 0 → (3 ± 7)/2 → 5 or −2.
Example 2: 4x² + 4x + 1 = 0 → discriminant 0 → x = −½.
Example 3: x² − 2x − 4 = 0 → (2 ± √20)/2 = 1 ± √5.
Common mistakes
- Sign errors with b: −b when b = −7 is +7.
- Dividing only √ by 2a instead of the whole numerator.
- Using the formula before setting the equation to zero.
- Squaring a negative b incorrectly: (−7)² = 49.
Try it yourself
- x² + 5x + 6 = 0
- 3x² − 2x − 1 = 0
- x² + 6x + 4 = 0 (exact)
- 2x² + x + 3 = 0
- A ball’s height is h = −5t² + 10t + 2. When does it land? (two decimals)
Answers: 1) −2, −3 2) 1, −⅓ 3) −3 ± √5 4) no real roots 5) t ≈ 2.18 s
Quick recap
- The quadratic formula solves any quadratic in standard form.
- Watch signs of the coefficients.
- The discriminant predicts the number of real roots.