ganak.appQuadratic Formula Calculator
Solve any quadratic equation (ax² + bx + c = 0). Find roots, discriminant, vertex, and parabola graph with steps. Real and complex roots.
Key Properties
Discriminant, Vieta's formulas & parabola
Equation Forms
Standard, vertex & factored representations
Parabola Graph
Visualize the parabola, roots, and vertex on the coordinate plane
Step-by-Step Solution
Detailed walkthrough of the quadratic formula applied to your equation
Identify coefficients
ax² + bx + c = 0
a = 1, b = -5, c = 6
x² - 5x + 6 = 0
Calculate the discriminant
Δ = b² − 4ac
Δ = (-5)² − 4(1)(6)
Δ = 25 − 24 = 1
Discriminant is positive
Δ > 0 → Two distinct real roots
Δ = 1 > 0
The equation has two distinct real roots.
Apply the quadratic formula
x = (−b ± √Δ) / 2a
x = (−(-5) ± √(1)) / (2 × 1)
x = (5 ± √(1)) / 2
Compute the roots
x₁ = (−b + √Δ) / 2a, x₂ = (−b − √Δ) / 2a
x₁ = (5 + 1) / 2, x₂ = (5 − 1) / 2
x₁ = 3, x₂ = 2
Verify with Vieta's formulas
x₁ + x₂ = −b/a, x₁ × x₂ = c/a
Sum = −(-5)/1 = 5, Product = 6/1 = 6
Sum of roots = 5, Product of roots = 6
What Is the Quadratic Formula?
The universal formula for solving quadratic equations
A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The quadratic formula provides a direct way to find the solutions (roots) of any quadratic equation, regardless of whether it can be factored.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
The expression under the square root, b² − 4ac, is called the discriminant (Δ). It determines the nature of the roots: positive means two distinct real roots, zero means one repeated root, and negative means two complex conjugate roots.
Understanding the Discriminant
How Δ = b² − 4ac determines root type
Δ > 0 — Two distinct real roots
The parabola crosses the x-axis at two points. Example: x² − 5x + 6 = 0 has Δ = 1, giving x = 2 and x = 3.
Δ = 0 — One repeated (double) root
The parabola touches the x-axis at exactly one point (the vertex). Example: x² − 6x + 9 = 0 has Δ = 0, giving x = 3 (repeated).
Δ < 0 — Two complex conjugate roots
The parabola does not cross the x-axis. Example: x² + 2x + 5 = 0 has Δ = −16, giving x = −1 ± 2i.
Vieta's Formulas & Equation Forms
Useful relationships between coefficients and roots
Vieta's formulas relate the coefficients of a quadratic to the sum and product of its roots without solving the equation:
x₁ + x₂ = −b/a
x₁ × x₂ = c/a
A quadratic equation can be expressed in three equivalent forms:
- Standard form: ax² + bx + c = 0 — directly shows coefficients.
- Vertex form: a(x − h)² + k = 0 — shows the vertex (h, k) of the parabola.
- Factored form: a(x − r₁)(x − r₂) = 0 — shows the roots directly (only when roots are real).
Common Mistakes to Avoid
Pitfalls students frequently encounter
- Forgetting to set the equation to zero — the formula only works when one side equals 0. Rearrange first.
- Sign errors with −b — when b is already negative, −b becomes positive. Use parentheses carefully.
- Dividing by 2 instead of 2a — the entire denominator is 2a, not just 2. If a = 3, divide by 6.
- Forgetting both ± solutions — the ± means you must compute both the + and − cases, giving two roots.
- Not simplifying radicals — always reduce √(b² − 4ac) to simplest form, e.g. √72 = 6√2.
Real-World Applications
Where quadratic equations appear in practice
- Projectile motion: The height of a thrown ball follows h(t) = −½gt² + v₀t + h₀. Solving for h = 0 gives the time of impact.
- Area optimization: Finding the dimensions of a rectangle with maximum area for a given perimeter involves solving a quadratic.
- Profit maximization: Revenue and cost models in economics often produce quadratic profit functions; the vertex gives maximum profit.
- Engineering: Parabolic shapes appear in satellite dishes, suspension bridge cables, and lens design, all modeled by quadratic equations.
Frequently Asked Questions
Common questions about quadratic equations and the quadratic formula
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Last updated Mar 2, 2026