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Quadratic Equation Solver & Graphing Helper

Discriminant and roots for ax^2+bx+c. Complex roots print as text when the discriminant is negative.

Page updated 2026-09-04.

Quadratic Equation Solver & Graphing Helper visual
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Default 1.

Default -3.

Default 2. Roots 1 and 2.

Calculated Results

Discriminant

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Vertex x

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Two real roots, found from the discriminant

For x² - 3x + 2 = 0 (a=1, b=-3, c=2), the roots are 2 and 1, with a discriminant of 1.00 and a vertex x-coordinate of 1.50.

The discriminant (b² - 4ac = (-3)² - 4(1)(2) = 9 - 8 = 1) determines the nature of the roots before you even solve for them: a positive discriminant (as here) means two distinct real roots exist, a discriminant of exactly zero means one repeated real root, and a negative discriminant means the roots are complex (not real) numbers.

The vertex x-coordinate (1.5, calculated as -b/(2a) = -(-3)/(2x1) = 1.5) sits exactly halfway between the two roots (1 and 2) -- this isn't a coincidence: a parabola is symmetric around its vertex, so its two roots (where it crosses the x-axis) are always equidistant from the vertex on either side.

What happens with zero or negative discriminants

Real roots when disc >= 0. Vertex x = -b/(2a). Not a graph renderer. A discriminant of exactly zero produces one repeated root rather than two distinct ones -- the parabola touches the x-axis at exactly one point (its vertex sits directly on the x-axis) rather than crossing through it at two separate points.

A negative discriminant means the quadratic formula would require taking the square root of a negative number, which produces complex (imaginary) roots rather than real ones -- graphically, this corresponds to a parabola that never crosses the x-axis at all, staying entirely above or entirely below it.

This solver reports the roots and key algebraic values (discriminant, vertex x) but doesn't render an actual visual graph of the parabola -- for a full visual sense of the curve's shape alongside these calculated values, a graphing tool would complement this numeric solver.

Related algebra and math tools

For a different structured math calculation involving matrices rather than a single equation, the Matrix Add/Multiply Calculator is a related tool.

For counting or combinatorics problems that come up alongside algebra coursework, the Combinations & Permutations Calculator covers that separate area of math.

Frequently Asked Questions (FAQ)

How are the roots 2 and 1 found for x² - 3x + 2 = 0?

Using the quadratic formula: x = (-b ± sqrt(discriminant)) / (2a) = (3 ± sqrt(1)) / 2 = (3 ± 1) / 2, giving x = 2 or x = 1.

What does the discriminant value tell you before solving for the roots?

Real roots when disc >= 0. Vertex x = -b/(2a). Not a graph renderer. A positive discriminant (like this equation's 1) means two distinct real roots exist. Zero means one repeated real root. A negative discriminant means the roots are complex, not real -- checking the discriminant's sign first tells you what kind of answer to expect.

Why does the vertex x-coordinate sit exactly between the two roots?

A parabola is symmetric around its vertex, so wherever it crosses the x-axis (its real roots), those crossing points are always equally spaced on either side of the vertex -- the vertex x-coordinate is mathematically the average of the two roots when real roots exist.

What happens if the discriminant is negative?

Real roots when disc >= 0. Vertex x = -b/(2a). Not a graph renderer. The equation has no real roots -- the quadratic formula would require the square root of a negative number, producing complex (imaginary) roots instead. Graphically, this means the parabola never touches the x-axis.

Does this tool show me a graph of the parabola?

No. Real roots when disc >= 0. Vertex x = -b/(2a). Not a graph renderer. It calculates the roots, discriminant, and vertex x-coordinate numerically, but doesn't render a visual plot of the curve -- a separate graphing tool would be needed to see the actual parabola shape.

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