What Is a Quadratic Equation?
A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0. The solutions — called roots — are the x-values that make the equation true. Every quadratic has exactly two roots (which may be equal, or complex numbers).
Method 1: Factoring
Factoring works best when the quadratic factors neatly into two binomials.
Example: Solve x² + 5x + 6 = 0
- Find two numbers that multiply to 6 and add to 5 → 2 and 3.
- Factor: (x + 2)(x + 3) = 0
- Set each factor to zero: x + 2 = 0 → x = −2; x + 3 = 0 → x = −3
Tip: Always check by substituting your answers back into the original equation.
Method 2: Completing the Square
This method works for any quadratic and is the foundation of the quadratic formula.
Example: Solve x² + 6x + 5 = 0
- Move the constant: x² + 6x = −5
- Add (b/2)² = (3)² = 9 to both sides: x² + 6x + 9 = 4
- Write as a perfect square: (x + 3)² = 4
- Take the square root: x + 3 = ±2
- Solve: x = −1 or x = −5
Method 3: The Quadratic Formula
The most universal method — works for any quadratic, no matter how messy the numbers.
The formula is: x = (−b ± √(b² − 4ac)) / 2a
Example: Solve 2x² − 4x − 6 = 0 (a=2, b=−4, c=−6)
- Calculate the discriminant: b² − 4ac = 16 + 48 = 64
- Apply the formula: x = (4 ± √64) / 4 = (4 ± 8) / 4
- Solutions: x = 12/4 = 3 or x = −4/4 = −1
The Discriminant: Predicting the Number of Solutions
| Discriminant (b² − 4ac) | Number of Real Roots | Nature |
|---|---|---|
| Positive | 2 | Two distinct real roots |
| Zero | 1 | One repeated real root |
| Negative | 0 | Two complex roots |
Method 4: Graphing
The roots of a quadratic are the x-intercepts of its parabola y = ax² + bx + c. While graphing is less precise by hand, it gives excellent visual intuition:
- If the parabola crosses the x-axis at two points → two real roots.
- If it touches the x-axis at one point → one repeated root (the vertex).
- If it doesn't touch the x-axis → no real roots (complex roots).
Choosing the Right Method
- Factoring: Use when coefficients are small integers and the equation factors easily.
- Completing the Square: Use to derive formulas or when you need vertex form.
- Quadratic Formula: Use always — it works for every case.
- Graphing: Use to estimate solutions or visualize the problem.
With practice, you'll instinctively know which method to reach for. Start with the quadratic formula as your safety net, and build toward faster factoring recognition over time.