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What is Quadratic Equation Solver?
A quadratic equation is any equation written in the form ax² + bx + c = 0, and solving it means finding the value or values of x that make it true. These equations turn up constantly in physics problems, geometry, and competitive exam mathematics. Rather than working through the formula by hand and risking an arithmetic slip, this solver takes your three coefficients and returns the roots — real or complex — immediately.
The Quadratic Formula
For any equation ax² + bx + c = 0, the roots are found using:
The expression under the square root, b² - 4ac, is called the discriminant. Its sign tells you what kind of roots to expect:
- Discriminant > 0 → two distinct real roots
- Discriminant = 0 → one repeated real root
- Discriminant < 0 → two complex (imaginary) roots
Worked Example
Take x² - 5x + 6 = 0, where a = 1, b = -5, c = 6.
Since the discriminant is positive, there are two real roots:
You can verify this by factoring: x² - 5x + 6 = (x - 3)(x - 2), confirming the roots are 3 and 2.
Now consider x² + 2x + 5 = 0, where a = 1, b = 2, c = 5. Here the discriminant = 2² - 4(1)(5) = 4 - 20 = -16, which is negative, so the roots are complex: x = -1 ± 2i.
How to Use This Calculator
- Enter the coefficient of x² in the a field.
- Enter the coefficient of x in the b field.
- Enter the constant term in the c field.
- Click Calculate to see both roots at once.
Need to double-check a root by raising it to a power? Our Power Calculator can help with that step.
Additional Worked Example: A Repeated Root
Take x² - 6x + 9 = 0, where a = 1, b = -6, c = 9. Discriminant = (-6)² - 4(1)(9) = 36 - 36 = 0. Since the discriminant is exactly zero, there is only one repeated real root: x = -(-6) / (2×1) = 6/2 = 3. This checks out by factoring too: x² - 6x + 9 = (x-3)(x-3) = (x-3)², confirming both roots coincide at x = 3. Graphically, this is the special case where the parabola's vertex just touches the x-axis at a single point instead of crossing through it twice, which is why this equation type is called a "perfect square trinomial."
Common Mistakes to Avoid
The most frequent slip is mismanaging the negative sign on b — remember the formula uses -b, so if b is already negative (like b = -5), then -b becomes positive 5, not -5. Students solving x² - 5x + 6 = 0 sometimes plug in b = -5 directly instead of computing -b = 5 first, flipping the sign of the entire answer. A second common error is forgetting the ± sign entirely and reporting only one root when the discriminant is positive — every positive-discriminant quadratic has exactly two distinct solutions, both of which should be checked by substituting back into the original equation. A third mistake is applying the quadratic formula when a = 0; without an x² term the equation is linear, not quadratic, and the formula's division by 2a becomes undefined. Finally, students often forget that a negative discriminant does not mean "no solution" — it means the solutions exist but are complex numbers, which is a different (and equally valid) mathematical answer rather than an error.
Why the Quadratic Formula Matters Beyond the Classroom
Quadratic equations model any situation where a quantity changes according to the square of another — which turns up far more often than most students expect. In physics, projectile motion (the height of a thrown ball over time) is a quadratic relationship, and solving for when the object hits the ground means solving a quadratic equation. In business and economics, profit-maximization and break-even problems are frequently modeled as quadratics, since revenue and cost curves often include squared terms. Engineers use quadratic equations when working with area-based design constraints, such as finding dimensions of a rectangular space given a fixed area and a required border. Board exam students across CBSE, ICSE, and state boards see quadratic equations as one of the highest-weightage chapters in Class 10 mathematics, making a reliable way to verify roots — and to understand what the discriminant reveals about a solution's nature — genuinely useful for exam preparation.
Frequently Asked Questions
Q: What does a negative discriminant mean for the roots of an equation?
A: A negative discriminant (b² − 4ac < 0) means the equation has no real number solutions — instead, it has two complex conjugate roots involving the imaginary unit i. Graphically, this means the parabola never crosses the x-axis at all, staying entirely above or below it.
Q: Why do some quadratics have only one root instead of two?
A: When the discriminant equals exactly zero, the ± in the quadratic formula collapses to a single value, producing one repeated root. Graphically, this happens when the parabola's vertex sits exactly on the x-axis, touching it at a single point rather than crossing it twice.
Q: How can I check my quadratic solution without the formula, just by factoring?
A: If the equation factors neatly, like x² − 5x + 6 = (x − 3)(x − 2), setting each bracket to zero directly gives the roots x = 3 and x = 2. Factoring only works cleanly for equations with "nice" integer roots, which is why the quadratic formula is needed as a universal backup method.
Q: What happens if I enter 0 for the "a" coefficient?
A: If a = 0, the x² term disappears and the equation is no longer quadratic — it becomes a simple linear equation, bx + c = 0, solvable by basic algebra instead. The quadratic formula requires a non-zero "a" value to remain valid.
Q: How are the two roots related to the original coefficients?
A: By Vieta's formulas, the sum of the two roots always equals −b/a, and their product always equals c/a — a handy way to sanity-check a solved equation without re-deriving it. For x² − 5x + 6 = 0, the roots 2 and 3 sum to 5 (matching −(−5)/1) and multiply to 6 (matching 6/1), confirming the answer is correct.
📅 Last reviewed: July 2026 · Formulas verified against RBI/SEBI/IT Dept guidelines.