Two distinct real roots
The parabola crosses the x-axis twice. If Δ is a perfect square and coefficients are integers, the roots are rational.
SOLVE · GRAPH · VERIFY
Quadratic Formula Calculator
Solve ax² + bx + c = 0 with real or complex roots, then inspect the discriminant, vertex, graph, exact form, completing-the-square work, and Vieta checks.
2 inputscoefficients or equation
3 methodsformula · square · factors
localno upload
QUADRATIC WORKBENCH
Enter the quadratic you actually have
Decimals and simple fractions such as 3/4 are accepted. a cannot be zero.
Supports quadratic, linear and constant terms on either side, e.g. 4x² + 3x - 7 = -4 - x. Parentheses are intentionally rejected rather than guessed.
STANDARD FORM—
VERTEX FORM—
FACTORED FORM—
EXACT ROOTS—
PARABOLA
Graph the equation
1×
Roots and vertex are marked when visible.
GEOMETRY & IDENTITIES
Read more than the roots
- y-intercept
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- Range
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- Focus
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- Directrix
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- Sum of roots
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- Product of roots
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- Enter a quadratic above.
DISCRIMINANT MAP
Know what Δ tells you before solving
One repeated real root
The vertex touches the x-axis. The repeated root is exactly the axis of symmetry, −b/(2a).
Two complex roots
The real parabola never reaches the x-axis. The solutions form a conjugate pair with equal real parts.
Fast structural check
For roots r₁ and r₂: r₁+r₂ = −b/a and r₁r₂ = c/a. This catches many sign mistakes.
DAILY QUADRATIC
One factorable challenge per UTC day
Same equation for everyone on the date. Solve it mentally, then check the pair of roots.
TODAY'S EQUATION—
Which ordered pair lists the two roots?
Choose an answer.
Method, precision & scope
Numerical stability
For two distinct real roots, the engine uses a cancellation-resistant variant of the quadratic formula when possible, then checks the values against the polynomial. Exact radical/fraction text is additionally shown for integer coefficients.
Equation parser
The text parser intentionally supports a transparent subset: x² terms, x terms, constants, decimals and simple fractions on either side of one equals sign. It does not pretend to understand arbitrary algebraic syntax or parentheses.
Graph
The canvas is a local visualization, not a symbolic graphing system. It marks real roots and the vertex and adapts its vertical scale to the sampled parabola.
Scope
This page solves one-variable quadratic equations with real numeric coefficients. It is an educational calculator, not a computer algebra system for higher-degree polynomials or symbolic parameters.