Quadratic Equation Solver — Solve ax² + bx + c = 0 Instantly
This free quadratic equation solver finds the solutions of equations in the form ax² + bx + c = 0. Enter the three coefficients and get both roots, along with the discriminant and solution type, instantly. Whether you're checking homework, studying for an exam, or solving real-world problems involving projectile motion or optimization, this tool provides accurate results with clear explanations.
Quadratic equations are among the most important types of equations in mathematics. They appear in physics, engineering, geometry, and countless real-world applications. Understanding how to solve them using the quadratic formula is a fundamental algebraic skill that opens the door to more advanced mathematical concepts.
Table of Contents
Solve ax² + bx + c = 0
1. What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation — the highest power of the variable x is 2. The name comes from the Latin word "quadratus," meaning square, because the variable is squared. Quadratic equations appear throughout algebra and are the gateway to understanding parabolas, projectile motion, and optimization problems.
The coefficient a cannot be zero — if it were, the x² term would disappear and the equation would become linear, not quadratic. The coefficients b and c can be any real numbers, including zero. When graphed, a quadratic equation produces a parabola — a U-shaped curve that opens upward when a is positive and downward when a is negative.
For related solvers, try our cubic equation solver or completing square calculator. For expanding quadratic expressions, see the expand binomials calculator.
2. The Quadratic Formula
To solve any quadratic equation, mathematicians use the quadratic formula — one of the most famous formulas in all of mathematics. It works for every quadratic equation, regardless of whether the roots are nice integers, messy fractions, or even complex numbers:
The ± symbol means there are generally two solutions — one using the plus sign and one using the minus sign. The expression under the square root, b² − 4ac, is called the discriminant. It determines how many real solutions the equation has and what type they are.
The entire formula can be derived from the standard form by a process called "completing the square," which is why it always works. See our completing square calculator for the derivation.
3. How the Calculator Works
The calculator applies the quadratic formula in four systematic steps:
Step 1: Read the coefficients
The calculator takes the values of a, b, and c from your input. It checks that a is not zero — if a = 0, the equation isn't quadratic and can't be solved by this method.
Step 2: Calculate the discriminant
Using the formula Δ = b² − 4ac, the calculator determines what's under the square root. This single number controls everything about the solutions.
Step 3: Determine the number of solutions
If Δ > 0: two real roots. If Δ = 0: one real root (repeated). If Δ < 0: no real solutions.
Step 4: Compute x using the quadratic formula
The calculator plugs the values into x = (−b ± √Δ) / (2a) and displays both solutions when they exist.
4. Types of Solutions
The discriminant (Δ = b² − 4ac) is the key to understanding quadratic solutions. It acts like a crystal ball — before you even solve the equation, the discriminant tells you what kind of answers to expect:
| Discriminant | Solutions | Graph | Example |
|---|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice | x² − 5x + 6 = 0 → x = 2, x = 3 |
| Δ = 0 | One repeated real root | Parabola touches x-axis at vertex | x² − 6x + 9 = 0 → x = 3 |
| Δ < 0 | No real solutions | Parabola never touches x-axis | x² + x + 1 = 0 → no real roots |
5. Worked Examples
Example 1: Two Real Solutions
Solve: x² − 5x + 6 = 0
Step 1: Identify a = 1, b = −5, c = 6.
Step 2: Calculate Δ = (−5)² − 4(1)(6) = 25 − 24 = 1.
Step 3: Δ > 0, so there are two real solutions.
Step 4: x = (5 ± √1) / 2 = (5 ± 1) / 2 → x = 2 or x = 3.
Example 2: One Repeated Solution
Solve: x² − 6x + 9 = 0
Step 1: a = 1, b = −6, c = 9.
Step 2: Δ = (−6)² − 4(1)(9) = 36 − 36 = 0.
Step 3: Δ = 0, so there's one repeated solution.
Step 4: x = 6 / 2 = x = 3 (twice).
Example 3: No Real Solutions
Solve: x² + x + 1 = 0
Step 1: a = 1, b = 1, c = 1.
Step 2: Δ = 1² − 4(1)(1) = 1 − 4 = −3.
Step 3: Δ < 0, so there are no real solutions.
6. Special Cases
Case 1: b = 0
The equation becomes ax² + c = 0. Solutions: x = ±√(−c/a). This works when −c/a ≥ 0.
Case 2: c = 0
The equation becomes ax² + bx = 0. Factoring gives x(ax + b) = 0, so x = 0 or x = −b/a.
Case 3: a = 1 (monic)
The equation is x² + bx + c = 0. The sum of the roots is −b and the product is c.
Case 4: Perfect square discriminant
When Δ is a perfect square (1, 4, 9, 16...), the roots are rational numbers.
7. Why Quadratic Equations Matter
Quadratic equations model countless real-world phenomena and appear across science, engineering, and mathematics:
🚀 Physics: Projectile motion — the path of a thrown ball follows a quadratic equation
🏗️ Engineering: Designing arches, bridges, and satellite dishes using parabolic shapes
📐 Geometry: Calculating areas and solving for unknown dimensions in Pythagorean problems
🎓 Education: Building algebraic foundation for calculus and higher mathematics
🔬 Biology: Modeling population growth with limited resources using quadratic equations
📊 Optimization: Finding maximum or minimum values in quadratic functions
8. Common Mistakes to Avoid
Mistake 1: Forgetting that a ≠ 0
If a = 0, the equation is linear (bx + c = 0), not quadratic. The quadratic formula involves division by 2a, which becomes division by zero.
Mistake 2: Misidentifying the sign of b in the formula
The formula has −b in the numerator. If b = −5, then −b = 5. A common error is forgetting the minus sign.
Mistake 3: Confusing the discriminant with the solution
The discriminant (b² − 4ac) tells you about the solutions but isn't the solution itself. You still need to take its square root.
Mistake 4: Forgetting the ± symbol
The quadratic formula always gives two values (unless Δ = 0). Forgetting the ± means finding only half the solutions.
Mistake 5: Incorrectly simplifying
Always simplify completely. x = (4 ± 2) / 2 gives x = 3 and x = 1, not x = (4 ± 2) / 2 left unsimplified.
9. Frequently Asked Questions
The discriminant (Δ = b² − 4ac) reveals everything about a quadratic's roots before you solve it. Δ > 0 means two real roots, Δ = 0 means one repeated real root, and Δ < 0 means no real roots. It also tells you if the roots are rational — if Δ is a perfect square, the roots will be rational numbers.
Yes. When the discriminant equals zero, the equation has exactly one real solution (called a repeated or double root). Geometrically, the parabola touches the x-axis at exactly one point — its vertex sits right on the axis.
The equation has no real solutions — the parabola doesn't cross the x-axis. In advanced mathematics, the solutions are complex numbers involving the imaginary unit i (where i² = −1). This calculator focuses on real solutions and will tell you when none exist.
The formula is derived by "completing the square" on the general form ax² + bx + c = 0. Starting with ax² + bx = −c, dividing by a, adding (b/2a)² to both sides, and taking the square root leads to the formula.
They all mean the same thing — the x-values that make the equation true. "Roots" is the algebraic term, "solutions" is the equation-solving term, and "zeros" is the graphing term (x-intercepts where y = 0).
The vertex is the highest or lowest point of the parabola, located at x = −b/(2a). The y-coordinate is found by plugging this x back into the equation. The vertex represents the maximum (if a < 0) or minimum (if a > 0) of the quadratic function.
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Conclusion
This quadratic equation solver is an essential math tool for solving polynomial equations quickly and accurately. It helps students understand algebra while saving time on calculations.
Whether you're checking homework, studying for an exam, or solving real-world problems involving projectile motion, optimization, or geometry, the quadratic formula is one of the most powerful tools in your mathematical toolkit. Bookmark this page for the next time you need to solve ax² + bx + c = 0 instantly.
Remember that while this tool is incredibly helpful, practicing quadratic formula applications by hand will deepen your understanding of algebra. Try solving simple quadratics mentally first, then verify with the calculator.
Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.