Quadratic Equation and the Quadratic Formula
The quadratic equation has the general form ax² + bx + c = 0 (a ≠ 0). Its solution is found with the quadratic formula.
Δ = b² − 4ac (discriminant) x = (−b ± √Δ) ÷ (2a) (quadratic formula) x₁ = (−b + √Δ) ÷ (2a) x₂ = (−b − √Δ) ÷ (2a)
The Discriminant (Δ)
- Δ > 0 → two distinct real roots
- Δ = 0 → one real root (double root)
- Δ < 0 → no real roots
Vertex of the Parabola
xᵥ = −b ÷ (2a) (axis of symmetry) yᵥ = −Δ ÷ (4a) (maximum or minimum value) a > 0 → parabola opens upward (minimum at yᵥ) a < 0 → parabola opens downward (maximum at yᵥ)
Vieta's Formulas
Sum and product of the roots without using the quadratic formula:
x₁ + x₂ = −b/a x₁ × x₂ = c/a
Frequently asked questions
What should I do if Δ < 0?
In the real numbers there is no solution. In the complex numbers the roots are imaginary (they involve i = √(−1)). In high school we say "there are no real roots".
Can the equation have only one root?
Yes! When Δ = 0, x₁ = x₂ = −b/(2a). This is called a double root or a root of multiplicity 2.
How do I check the roots I found?
Substitute each root into the original equation. The result must be 0. Example: for x=2 in x²−5x+6: 4−10+6 = 0 ✓.
Why is a ≠ 0 in a quadratic equation?
If a = 0, the equation becomes bx + c = 0, which is first degree (linear). A quadratic equation requires the x² term.