Enter the coefficients for ax² + bx + c = 0. All three are required (a cannot be zero).
__ x² + __ x + __ = 0
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Quadratic Equation FAQ
Everything you need to know about solving quadratic equations.
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It solves any equation in the standard form ax² + bx + c = 0. The ± sign means there are usually two solutions — one with addition and one with subtraction.
The discriminant is b² − 4ac. It tells you the nature of the roots before you solve:
- Positive → two distinct real roots
- Zero → exactly one real root (the parabola touches the x-axis)
- Negative → two complex (imaginary) roots (parabola doesn't cross x-axis)
With a=1, b=5, c=6:
Discriminant = 5² − 4(1)(6) = 25 − 24 = 1
x = (−5 ± √1) / 2 = (−5 ± 1) / 2
x₁ = (−5 + 1)/2 = −2
x₂ = (−5 − 1)/2 = −3
Check: (x+2)(x+3) = x² + 5x + 6 ✓
Discriminant = 5² − 4(1)(6) = 25 − 24 = 1
x = (−5 ± √1) / 2 = (−5 ± 1) / 2
x₁ = (−5 + 1)/2 = −2
x₂ = (−5 − 1)/2 = −3
Check: (x+2)(x+3) = x² + 5x + 6 ✓
When the discriminant is negative (b² − 4ac < 0), you'd need to take the square root of a negative number. This gives complex numbers involving i (where i = √−1). The equation has two complex conjugate roots like 3 + 2i and 3 − 2i. Graphically, the parabola never crosses the x-axis.
For f(x) = ax² + bx + c, the vertex is at:
x = −b / (2a)
Then substitute that x back into the equation to find y.
If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
x = −b / (2a)
Then substitute that x back into the equation to find y.
If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.