Quadratic equation calculator guide
A quadratic equation in standard form is ax² + bx + c = 0 with a nonzero leading coefficient. The quadratic formula works whether the expression factors conveniently or not.
The discriminant b² - 4ac determines the root type. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives a complex-conjugate pair.
How to use the quadratic formula calculator
- Write standard form: Move every term to one side so the other side is zero.
- Enter a, b, and c: Include signs and use a nonzero value for a.
- Solve: Select Solve quadratic to calculate roots and parabola properties.
- Read the root type: Use the discriminant and root classification to distinguish real and complex results.
Formula and variables
The quadratic formula solves the standard-form equation. Evaluating the polynomial at x = -b/(2a) gives the vertex y-coordinate.
x = (-b +/- sqrt(b^2 - 4ac)) / (2a); vertex x = -b/(2a)- a — Leading coefficient
- Nonzero coefficient of x squared; its sign determines the opening direction.
- b — Linear coefficient
- Coefficient of x.
- c — Constant term
- Also the y-intercept of the associated parabola.
- D — Discriminant
- D = b squared minus 4ac, which classifies the roots.
Solve x² - 3x - 4 = 0
Find the roots and vertex of the quadratic.
- a
- 1
- b
- -3
- c
- -4
- D = (-3)^2 - 4(1)(-4) = 25
- x = (3 +/- 5) / 2
- Vertex x = 3/2; substitute to get y = -25/4
Result: The roots are -1 and 4, and the vertex is (1.5, -6.25).
The positive discriminant gives two x-intercepts, and positive a means the parabola opens upward.
Understanding your results
Roots
Real roots are x-intercepts. A complex-conjugate pair means the real graph does not cross the x-axis.
Vertex and axis
The axis x = -b/(2a) passes through the vertex, the minimum point when a is positive and maximum point when a is negative.
Assumptions
- Coefficients are finite real numbers.
- Coefficient a is nonzero.
- Complex results are reported in rectangular form with a nonnegative magnitude before plus/minus interpretation.
Limitations
- The calculator does not factor symbolic expressions.
- It does not accept equations outside standard coefficient form.
- Floating-point arithmetic can round irrational roots.
- Very large coefficients can exceed the numeric range even when a scaled symbolic form would be manageable.
Common mistakes
- Forgetting that a must be nonzero.
- Dropping a negative sign when substituting b.
- Using b squared minus 4c instead of b squared minus 4ac.
- Dividing only the square-root term by 2a.
- Assuming a negative discriminant means no roots instead of no real roots.
Practical use cases
Equation solving
Find all roots even when a quadratic does not factor cleanly over integers.
Parabola analysis
Read the vertex, axis, intercept, and opening direction for graphing or optimization.
Planning and decision guide
Keep coefficient signs explicit
Place parentheses around negative coefficient values during hand substitution, and delay rounding until the final display or application step.
Frequently asked questions
What does the discriminant tell me?
Positive gives two real roots, zero gives one repeated real root, and negative gives two complex-conjugate roots.
Can a quadratic have only one root?
It can have one distinct real value with multiplicity two when the discriminant is zero.
Why must a be nonzero?
If a is zero, the x-squared term disappears and the equation is linear rather than quadratic.
How is the vertex calculated?
Its x-coordinate is -b divided by 2a; substitute that value into the polynomial for the y-coordinate.
Why are complex roots paired?
A quadratic with real coefficients has nonreal roots in conjugate pairs with equal real parts and opposite imaginary parts.
Sources and review
- Quadratic Equations — OpenStax College Algebra 2e. Accessed 2026-08-29.
- Chapter 2 Key Concepts — OpenStax College Algebra 2e. Accessed 2026-08-29.
Reviewed 2026-08-29.