Quadratic Equation Calculator
Solve equations of the form ax² + bx + c = 0
What Is the Quadratic Equation Calculator?
This calculator solves quadratic equations of the form ax² + bx + c = 0, finding both roots (solutions) using the quadratic formula, and tells you whether the roots are real or complex based on the discriminant. A quadratic equation is any polynomial equation of degree 2 — the highest power of x is exactly 2 — and its graph is always a parabola.
The quadratic formula itself has a long history: methods for solving quadratic-type problems appear in Babylonian mathematics from around 2000 BC, and various partial solutions were developed independently across ancient Greek, Indian, and Islamic mathematics over the following centuries. The general symbolic form used today, covering negative and complex roots consistently, wasn't fully standardized until algebraic notation matured in the 16th and 17th centuries.
If your equation's x² term has a zero coefficient, it isn't quadratic at all — see the Linear Equation Solver instead. And if you just need to evaluate ax² + bx + c at a specific x value rather than solve for where it equals zero, the Polynomial Calculator does that more directly.
Quadratic Equation Calculator Formula
x = (−b ± √(b² − 4ac)) / 2a
How Is the Quadratic Equation Calculator Calculated?
The discriminant (b² − 4ac) inside the square root determines how many real solutions exist: positive gives two distinct real roots, zero gives one repeated real root (the parabola just touches the x-axis), and negative means the parabola never crosses the x-axis, giving two complex roots instead.
The formula itself comes from a technique called "completing the square," which rewrites ax² + bx + c = 0 so that the x terms form a perfect square, then isolates x by taking a square root of both sides. That's also exactly why the discriminant sits under a square root in the final formula — it's the leftover expression from that same square-completing step, and its sign directly determines whether that square root produces a real or an imaginary number.
Quadratic Equation Calculator Example
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): D = 25 − 24 = 1, so roots are x₁ = 3 and x₂ = 2.
For 2x² − 3x − 2 = 0 (a=2, b=−3, c=−2): D = 9 − (4×2×−2) = 9 + 16 = 25, so roots are x₁ = 2 and x₂ = −0.5.
For x² + 2x + 5 = 0 (a=1, b=2, c=5): D = 4 − 20 = −16, a negative discriminant, giving complex roots x₁ = −1 + 2i and x₂ = −1 − 2i.
How to Use the Quadratic Equation Calculator
Step 1
Enter the coefficients a, b, and c from your equation.
Step 2
Click Solve Equation.
Step 3
View the discriminant, nature of the roots, both solutions (x₁ and x₂), and the parabola graph.
Step 4
Check the discriminant's sign before reading the roots, so the real vs. complex result isn't a surprise.
Step 5
Try flipping the sign of a to see how the parabola's direction (and the graph) changes while the discriminant stays the same.
Benefits
- Packages your solved roots into a branded image card, useful for sharing a worked answer with a study group.
- Handles all three root cases: real & distinct, real & equal, and complex.
- Visualizes the parabola with root markers for real solutions.
- Shows the discriminant so you understand why the roots turned out the way they did.
- Saves you from a completing-the-square derivation every time you just need the two roots.
- Free and instant, with no account or software installation required.
Common Quadratic Equation Calculator Scenarios
Scenario 1
Solving algebra homework or exam problems.
Scenario 2
Physics problems involving projectile motion (a parabolic trajectory).
Scenario 3
Optimization problems where a parabola models cost, profit, or area.
Scenario 4
Finding the break-even points of a quadratic cost or revenue model.
Scenario 5
Checking whether a parabola in a given problem actually crosses the x-axis before solving further.
Scenario 6
Verifying factored forms of a quadratic by comparing them to the calculated roots.
Understanding Your Result
The roots (x₁, x₂) are the x-values where the parabola y = ax² + bx + c crosses (or touches) the x-axis. For complex roots, the parabola never touches the x-axis at all — the graph stays entirely above or below it.
The sign of a also matters for interpreting the graph: a positive a means the parabola opens upward, so a negative discriminant means the whole curve sits above the x-axis; a negative a means it opens downward, so a negative discriminant means the curve sits entirely below the x-axis. Either way, the vertex (the parabola's highest or lowest point) sits exactly halfway between the two roots when real roots exist.
Tips
- Check the discriminant sign first to know what kind of roots to expect before solving.
- If a is negative, the parabola opens downward instead of upward.
- Use the graph to visually confirm where the curve crosses zero.
- If both roots come out equal, double-check whether your equation is actually a perfect square trinomial.
- For quick mental estimates, the sum of the roots always equals −b/a and the product always equals c/a — handy for checking your calculator result by hand.
Common Mistakes
- Entering a = 0, which turns the equation linear rather than quadratic.
- Forgetting the ± sign, which produces only one of the two roots.
- Mixing up b² with (b)² when b is negative — squaring always gives a positive result.
- Misreading a negative discriminant as "no solution" rather than "two complex solutions" — the equation still has roots, just not real ones.
- Dividing only part of the numerator by 2a instead of the entire (−b ± √D) expression.
Frequently Asked Questions
What does the discriminant tell me?
The discriminant (b² − 4ac) determines the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.
Why must a not be zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic — use the Linear Equation Solver for that case instead, since the quadratic formula divides by 2a and is undefined when a is 0.
What are complex roots?
When the discriminant is negative, the roots involve the imaginary unit i (√-1) — these are valid mathematical solutions but not real numbers, and they always come in a conjugate pair (real ± imaginary·i).
What does the graph show?
The graph plots y = ax² + bx + c around its vertex, with red markers showing where the curve crosses the x-axis (the real roots), if any.
Can a quadratic equation have only one solution?
Yes, when the discriminant equals zero, both roots are equal, meaning the parabola touches the x-axis at exactly one point — this is sometimes called a repeated or double root.
What does it mean if the discriminant is negative?
A negative discriminant means the equation has no real roots — instead, it has two complex (imaginary) roots, since you'd be taking the square root of a negative number.
Can a quadratic equation have exactly one solution?
Yes — when the discriminant equals exactly zero, both roots are identical, giving a single repeated real solution (sometimes called a 'double root').
Why must the coefficient a not be zero?
If a were zero, the x² term would disappear, turning the equation into a linear equation instead of a quadratic one — the quadratic formula requires dividing by 2a, which is undefined if a is zero.
What does the graph of a quadratic equation look like?
It forms a parabola — a U-shaped curve that opens upward if a is positive or downward if a is negative, and the roots are where that curve crosses the x-axis.
Can I share my quadratic equation solution as an image?
Yes — tap Share and, on supported devices, your roots are shared as a branded image card, not just a text link.
How do I know the number of real roots without fully solving the equation?
Just calculate the discriminant (b² − 4ac) — its sign alone tells you whether there are two, one, or zero real roots, without needing to compute the roots themselves.
Is there a shortcut to check my roots without redoing the full quadratic formula?
Yes — Vieta's formulas state that the sum of the roots always equals −b/a and their product always equals c/a, which is a quick way to sanity-check a calculated pair of roots by hand.
How old is the quadratic formula?
Methods for solving quadratic-type problems date back to Babylonian mathematics around 2000 BC, though the fully general symbolic formula used today wasn't standardized until algebraic notation matured in the 16th and 17th centuries.
References
Important Information
Complex roots are shown in the form real ± imaginary·i, but are not plotted on the graph.
Last updated: July 25, 2026