A quadratic equation is any equation that can be written in the standard form:
ax² + bx + c = 0
where a, b, and c are real numbers and a ≠ 0. These equations appear everywhere — from physics (projectile motion) to finance (profit optimization) to engineering (signal processing).
Method 1: Factoring
Factoring works best when the equation breaks neatly into two binomials.
Example: Solve x² − 5x + 6 = 0
- Find two numbers that multiply to 6 (the constant) and add to −5 (the coefficient of x)
- Those numbers are −2 and −3 (because −2 × −3 = 6 and −2 + −3 = −5)
- Write: (x − 2)(x − 3) = 0
- Set each factor to zero: x = 2 or x = 3
When to Use Factoring
- The equation has integer roots
- You can easily spot the factor pairs
- Quick mental math is possible
Method 2: Completing the Square
This method works for any quadratic and is the basis for deriving the quadratic formula.
Example: Solve x² + 6x + 2 = 0
- Move the constant: x² + 6x = −2
- Take half the coefficient of x (which is 6/2 = 3), square it (3² = 9)
- Add 9 to both sides: x² + 6x + 9 = 7
- Factor the left side: (x + 3)² = 7
- Take the square root: x + 3 = ±√7
- Solve: x = −3 ± √7 ≈ −0.354 or −5.646
Method 3: The Quadratic Formula
The most powerful method — it works for every quadratic equation:
x = (−b ± √(b² − 4ac)) / 2a
Example: Solve 2x² + 3x − 5 = 0
Here, a = 2, b = 3, c = −5:
- Calculate the discriminant: Δ = b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49
- Since Δ > 0, there are two real roots
- x = (−3 ± √49) / (2 × 2) = (−3 ± 7) / 4
- x₁ = (−3 + 7) / 4 = 1
- x₂ = (−3 − 7) / 4 = −2.5
Understanding the Discriminant
The discriminant Δ = b² − 4ac tells you what kind of roots to expect:
| Discriminant | Roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | Two complex (imaginary) roots |
Tips for Success
- Always write in standard form first (ax² + bx + c = 0)
- Check your answers by substituting back into the original equation
- Try factoring first — it's the fastest when it works
- Use the formula when factoring isn't obvious
- Don't forget that "no real solutions" is a valid answer when Δ < 0
Practice Problems
Try these on your own, then use our Quadratic Solver to check:
- x² − 7x + 12 = 0
- 3x² + 2x − 1 = 0
- x² + 4x + 4 = 0
- x² + x + 1 = 0 (hint: check the discriminant!)
Ready to solve quadratic equations instantly? Try our interactive Quadratic Solver — it shows step-by-step solutions, vertex form, and even plots the parabola.