Quadratic Equation Solver
NewSolve ax² + bx + c = 0 using the quadratic formula, showing the discriminant, both roots (real or complex), and the parabola's vertex.
1x² + -3x + 2 = 0
Roots are computed with the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two complex conjugate roots (shown as a ± bi).
Frequently Asked Questions
What does the discriminant tell me about the roots?
A positive discriminant (b² − 4ac > 0) means two distinct real roots; zero means exactly one repeated real root (the parabola just touches the x-axis); a negative discriminant means no real roots, only two complex conjugate roots.
What do the complex roots shown as "a + bi" mean?
When the discriminant is negative, the square root becomes imaginary, so the two roots take the form (real part) ± (imaginary part)i — this means the parabola never crosses the x-axis, and the roots only exist in the complex number plane.
What is the vertex, and why is it shown?
The vertex (−b/2a, f(−b/2a)) is the turning point of the parabola — its minimum if a is positive or maximum if a is negative — useful for graphing or understanding the equation's overall shape beyond just its roots.
Why can't coefficient a be 0?
If a = 0, the x² term disappears entirely and the equation becomes linear (bx + c = 0), which has at most one root and isn't solvable with the quadratic formula (it would involve dividing by zero in the 2a denominator).
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