Quadratic Formula Calculator
Solve ax2 + bx + c = 0 step by step: exact radical roots, decimals, complex pairs, and the vertex.
ax2 + bx + c = 0
Answer:
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How to use the quadratic formula calculator
- Enter a, b, and c from ax^2 + bx + c = 0. Signs matter.
- Read both roots in exact radical form and as decimals.
- Follow the discriminant, substitution, and vertex steps.
What it is for
The quadratic formula is memorized but rarely unpacked. This tool shows the discriminant decision, the substitution, simplified radicals instead of messy decimals alone, and the vertex, which is what graphing questions actually ask about.
How it works
The discriminant b^2 - 4ac is computed exactly with floating point, then reduced: its square part is pulled out of the radical, and the whole root is reduced by the greatest common divisor. Negative discriminants produce the complex conjugate pair (-b ± √(4ac-b²)·i)/(2a). The vertex uses (-b/2a, -Δ/4a).
Reading the discriminant: a worked example
For x² - 5x + 6 = 0, the discriminant is 25 - 24 = 1. Positive means two real roots: (5 ± 1)/2 gives x = 3 and x = 2, and indeed (x - 3)(x - 2) expands back to the original. Zero would mean one repeated root at the vertex, and negative a complex pair - so computing Δ first tells you which answer shape to expect before any algebra.