Quadratic Formula Calculator
Enter the coefficients a, b, and c of a quadratic equation and instantly find both roots, the discriminant, and whether the roots are real or complex. The quadratic formula solves every second-degree polynomial equation.
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Formula
x = (−b ± √(b² − 4ac)) / 2a
The quadratic formula gives both solutions of ax² + bx + c = 0. The term under the square root — b² − 4ac — is called the discriminant. If it is positive, there are two distinct real roots. If it equals zero, there is exactly one real root (the parabola touches the x-axis at one point). If negative, the roots are complex (the parabola doesn't cross the x-axis). The ± sign means you compute the formula twice: once adding the square root and once subtracting it.
How to use the Quadratic Formula Calculator
- 1
Enter your coefficient a
Coefficient of x²
- 2
Enter your coefficient b
Coefficient of x
- 3
Enter your coefficient c
Constant term
- 4
Read your results instantly
Results update in real time as you type.
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Understanding the discriminant
The discriminant b² − 4ac is the most informative single number you can compute from a quadratic. Before finding the roots, check its sign:
If b² − 4ac > 0: the equation has two distinct real roots. The parabola crosses the x-axis at two points. A large positive discriminant means the roots are far apart; a small one means they're close together.
If b² − 4ac = 0: there is exactly one real root, called a repeated root or double root. The parabola is tangent to the x-axis — it just touches without crossing.
If b² − 4ac < 0: there are no real roots. The parabola sits entirely above or below the x-axis. The roots are complex conjugates: (−b ± i√|Δ|) / 2a, where i is the imaginary unit.
Where quadratic equations appear
Quadratic equations govern any system where a quantity depends on the square of another. In physics, projectile motion follows y = y₀ + v₀t − ½gt², a quadratic in time. Setting y = 0 and solving tells you exactly when an object hits the ground.
In economics, profit functions are often quadratic: revenue is linear, but costs may include a quadratic component that causes diminishing returns. The vertex of the parabola gives the profit-maximizing output level.
In engineering, the resonant frequency of a circuit, the stress on a beam, and the trajectory of a lens ray all involve quadratic relationships. Solving for the zeros — the roots — answers questions like "at what angles does light emerge?" or "at what voltage does the circuit saturate?"
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Alternative methods for factoring quadratics
The quadratic formula always works, but when coefficients are small integers, factoring by inspection is faster. For x² − 5x + 6, look for two numbers that multiply to 6 and add to −5: those are −2 and −3. So the factored form is (x − 2)(x − 3), giving roots x = 2 and x = 3.
Completing the square is a third approach that also derives the quadratic formula itself. Rewrite ax² + bx + c = 0 as a(x + b/2a)² = (b² − 4ac)/4a, then isolate x.
For numerical roots without exact forms, Newton's method converges quadratically: guess x₀, then iterate xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Two or three steps typically give ten significant digits.
Tips & Insights
Check your roots by substituting back
After finding x₁ and x₂, substitute each back into ax² + bx + c and verify the result is zero (or very close to zero — floating-point arithmetic may give something like 1.4e-14 instead of exactly 0). If the result is large, re-check your signs for b and c.
The sum and product of roots
By Vieta's formulas, x₁ + x₂ = −b/a and x₁ × x₂ = c/a — without solving the quadratic at all. These relations let you verify roots quickly: for x² − 5x + 6, the sum should be 5 and the product 6. Roots 2 and 3: 2 + 3 = 5 ✓, 2 × 3 = 6 ✓.
Negative leading coefficient flips the parabola
When a < 0, the parabola opens downward. The roots still represent x-axis crossings, but the vertex is now a maximum instead of a minimum. This matters in optimization problems: maximizing profit or throwing range both involve downward parabolas.
Worked Examples
Finding when a ball hits the ground
Discriminant = 400. Roots: x = 0 s (launch) and x = 4 s (landing). The ball is in the air for exactly 4 seconds.
Solving a classroom quadratic
Discriminant = 25 − 24 = 1. Root 1: (5 + 1)/2 = 3. Root 2: (5 − 1)/2 = 2. The equation x² − 5x + 6 = 0 factors as (x − 2)(x − 3).
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Frequently Asked Questions
What does the discriminant tell me?
The discriminant b² − 4ac tells you how many real solutions exist. Positive: two real roots. Zero: one repeated real root. Negative: no real roots (two complex roots).
Can I use the quadratic formula when a = 0?
No — when a = 0, the equation becomes linear (bx + c = 0), which has one solution x = −c/b. The formula requires a ≠ 0.
What are complex roots?
When the discriminant is negative, the square root produces an imaginary number. The two roots become complex conjugates of the form p + qi and p − qi, where i = √(−1). They don't correspond to points on the real number line.
How do I write the equation in standard form?
Standard form is ax² + bx + c = 0. Expand and collect all terms on one side before identifying a, b, and c. For 3x² = 2x − 5, rearrange to 3x² − 2x + 5 = 0, giving a = 3, b = −2, c = 5.
What is a double root?
A double root occurs when the discriminant equals zero. The formula gives the same value for both ± cases: x = −b / 2a. Geometrically, the parabola is tangent to the x-axis at exactly one point.
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