Quadratic Formula Calculator

Solve quadratic equations ax² + bx + c = 0

Check Quadratic Formula Calculator

1x² +-5x +6 = 0
Examples:

Formula result

Copy-ready result handoff

Use this after solving quadratic roots and reviewing the discriminant.

Common searches
quadratic formula calculatorquadratic rootsdiscriminant
Input check

Confirm coefficients a, b, and c match ax^2 + bx + c = 0.

Assumption

The quadratic formula applies when a is not zero.

Quadratic solution note

Copy this after replacing the bracketed result with the value shown by the calculator.

Quadratic result: [paste roots, discriminant, and real or complex status].
Inputs checked: Confirm coefficients a, b, and c match ax^2 + bx + c = 0.
Assumption: The quadratic formula applies when a is not zero.
Use case: Algebra homework, graph intercept check, or root verification.
Next check: Use Percentage Calculator when comparing root movement or relative change.

Enter coefficients a, b, and c for ax^2 + bx + c = 0, then check roots, discriminant, vertex, and graph behavior together. The discriminant tells you what kind of roots to expect before you read the final answer.

How the result is built

Formula, Discriminant, And Worked Example

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
D = b^2 - 4ac

D greater than 0 gives two real roots, D equal to 0 gives one repeated root, and D less than 0 gives a complex-conjugate pair.

Worked example

For x^2 - 5x + 6 = 0, D = 25 - 24 = 1, so x = (5 +/- 1) / 2. The roots are x = 3 and x = 2.

Assumptions and limits
  • The coefficient a must not be zero; otherwise the equation is linear.
  • The equation should be rearranged into standard form before entering coefficients.
  • The graph is a numeric visualization, not a symbolic proof of every algebra step.
Common mistakes
  • Forgetting the negative sign in -b.
  • Calculating b^2 - 4ac with the wrong sign for c.
  • Entering coefficients before moving all terms to one side of the equation.

Quadratic Formula, Roots, And Graph Checks

Calculator Task Context

Use this calculator to solve equations in the form ax^2 + bx + c = 0 and inspect the discriminant.

Formula And Method Used

The calculator applies x = (-b +/- sqrt(b^2 - 4ac)) / (2a), then classifies the result by the discriminant value.

Worked Example

  1. For x^2 - 5x + 6 = 0, a = 1, b = -5, c = 6.
  2. The discriminant is 25 - 24 = 1, so x = (5 +/- 1)/2 = 3 and 2.

Common Mistakes And Limits

  • The coefficient a cannot be zero; otherwise the equation is linear, not quadratic.
  • Keep the sign of b when substituting into -b.
  • A negative discriminant means complex roots, not no algebraic roots.

Related Calculators

Root CalculatorUse it to verify the discriminant square root.Scientific CalculatorUse it for manual substitution checks.Factor CalculatorUse factoring when the quadratic splits into simple integer factors.

How to use the quadratic formula calculator

Rewrite the equation as ax^2 + bx + c = 0, then enter the three coefficients exactly as signed numbers. Read the discriminant first to confirm whether the roots should be real, repeated, or complex.

Roots versus graph behavior

Real roots are x-intercepts. A repeated root touches the x-axis at the vertex. Complex roots mean the parabola does not cross the x-axis, even though the equation still has two algebraic solutions.

Related tools

Use the Root Calculator for radical checks, the Scientific Calculator for supporting arithmetic, and the Exponent Calculator when powers are the main source of error.