Can I enter a full equation instead of only a, b, and c?

Yes. Enter both sides of the equation; the solver expands parentheses, combines like terms, and finds the coefficients for the quadratic formula.

What if the quadratic coefficient is zero?

Then the squared term disappears and the equation becomes linear. The solver reports a linear root, an identity, or a contradiction.

What does a complex root mean?

When the discriminant is negative, there are no real roots, but there is a pair of complex conjugate roots. The graph then has no real x-intercepts.

Why does the calculator show excluded values?

Excluded values appear when x is in a denominator. Any root that makes an original denominator zero is extraneous and must be rejected.

Why does the solver reject degree higher than 2?

After expansion, the calculator checks the maximum degree of x. A degree higher than 2 is not a quadratic equation.

Why show a parabola graph?

The graph helps verify the number and location of real roots, the vertex, and the opening direction of the quadratic function.

Quadratic Equation Solver

Solve quadratic equations from a full equation or standard coefficients. See the discriminant, real or complex roots, Vieta checks, excluded values, and a parabola graph.

Enter a quadratic equation

Supports parentheses, multiplication, division, squared x terms, and simple fractional equations. The simplified degree must be no higher than 2.

Quick input

Examples

Discriminant D

Positive: two distinct real roots

First root

Second root

Vieta check

Root sum: 5

Root product: 6

Step-by-step solution

The input equation was parsed.

Move all terms to one side and collect coefficients.

quadratic coefficient: 1, linear coefficient: -5, constant term: 6

The discriminant is 1.

The discriminant is positive, so there are two distinct real roots: 2 and 3.

By Vieta's formulas, the root sum is 5 and the product is 6.

Parabola graph

Vertex: (2.5, -0.25). The parabola opens upward.

Quadratic Equation Solver accepts a full equation, reduces it to canonical form, expands parentheses and squares, clears supported denominators, and marks extraneous roots when the original equation excludes them.

What this quadratic equation solver handles

English search intent for a quadratic equation solver expects more than a three-coefficient form. The calculator accepts a full equation, expands parentheses, moves terms to one side, reduces the expression to standard form, and then solves the quadratic or a linear special case.

a x^{2} + b x + c = 0

a is the quadratic coefficient, b is the linear coefficient, and c is the constant term after simplification. For a true quadratic equation, a is not zero.

Enter a standard equation such as - 5x +.
Use parentheses, products of linear factors, and squared binomials.
Let the solver collect coefficients and compute the discriminant.
Review excluded values when the original equation has x in a denominator.

Discriminant and quadratic formula

After reducing the equation, the solver computes the discriminant and classifies the roots as two real roots, one repeated real root, or a complex conjugate pair.

D = b^{2} - 4 a c

D is the discriminant; a, b, and c come from the simplified standard form.

x_{1, 2} = \frac{- b \pm D}{2 a}

x1 and x2 are the two real roots when D is positive.

x = \frac{- b}{2 a}

x is the repeated root when D equals zero.

x_{1, 2} = \frac{- b \pm i ∣ D ∣}{2 a}

i is the imaginary unit; this form is used when D is negative.

Discriminant	Root type	What the calculator shows
	Two distinct real roots	Both roots and a Vieta check
	One repeated real root	The double root and tangent point on the x-axis
	Two complex conjugate roots	The complex pair and a graph with no real x-intercepts

Parentheses, fractions, and excluded values

The solver supports more than coefficient boxes. It can parse parentheses, squared binomials, products of linear factors, and simple fractional equations as long as clearing denominators still leaves degree 2 or lower.

(x + 1)^{2} = 4

(x - 1) (x + 3) = 0

\frac{x ^{2} - 1}{x - 1} = 3

Excluded values and extraneous roots

If a denominator contains x, the calculator records values that would make the original denominator zero. A calculated root that violates the original domain is rejected as extraneous.

Vieta check and parabola graph

For two real roots, the calculator shows the sum and product check. The graph helps connect the roots with x-intercepts, the vertex, and the direction the parabola opens.

x_{1} + x_{2} = - \frac{b}{a}

x1 and x2 are the roots; their sum checks against the simplified coefficients.

x_{1} x_{2} = \frac{c}{a}

x1 and x2 are the roots; their product checks against the constant term c.

y = a x^{2} + b x + c

y is the value of the quadratic function shown as a parabola.

(- \frac{b}{2 a}, - \frac{D}{4 a})

a, b, and D determine the vertex coordinates after simplification.

When the equation is not quadratic

The page is built for one-variable equations that simplify to degree 2 or lower. If the quadratic term disappears, the solver reports the linear case. If the degree becomes higher, use a more specific algebra solver.

Situation	Meaning	Next step
No squared term remains	The equation is linear	Use the result here or open the linear equation solver
A cubic term appears	The equation is not quadratic	Use the cubic equation solver
Only even fourth-degree terms appear	This is a biquadratic pattern	Use the biquadratic equation solver
x appears in an exponent	This is an exponential equation	Use a different solver

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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