Quadratic Formula Calculator

What Is the Quadratic Formula?

The quadratic formula solves any equation written in the standard form ax² + bx + c = 0, where a, b, and c are real number coefficients and a ≠ 0. The formula states:

x = (−b ± √(b² − 4ac)) / (2a)

The ± symbol means the formula produces two solutions simultaneously. The first root uses the plus sign (+√), and the second root uses the minus sign (−√). As detailed by Khan Academy's quadratic formula article, this single expression handles all quadratic equations regardless of whether the roots are real, repeated, or complex.

Understanding the Variables

• a — Leading Coefficient: The coefficient of the x² term. Must be non-zero; if a = 0, the equation is linear, not quadratic. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).
• b — Linear Coefficient: The coefficient of the x term. Controls the horizontal position of the parabola's vertex. When b = 0, the parabola is symmetric about the y-axis.
• c — Constant Term: The y-intercept of the parabola. The curve always passes through the point (0, c).

Deriving the Formula: Completing the Square

The quadratic formula is derived by applying the completing the square technique to the general equation ax² + bx + c = 0. The derivation proceeds as follows:

• Divide every term by a: x² + (b/a)x + c/a = 0
• Isolate the constant: x² + (b/a)x = −c/a
• Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b² − 4ac) / (4a²)
• Factor the left side as a perfect square: (x + b/2a)² = (b² − 4ac) / (4a²)
• Take the square root and solve for x: x = (−b ± √(b² − 4ac)) / (2a)

This derivation, covered in depth by the University of Wisconsin-Whitewater's Methods for Solving Quadratic Equations, demonstrates why the formula works universally — it encodes the completing-the-square process into a single reusable expression applicable to every quadratic equation with real or complex coefficients.

The Discriminant: Predicting Root Behavior

The expression under the radical sign, Δ = b² − 4ac, is called the discriminant. Its value reveals the nature of the roots before performing the full calculation:

• Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two separate points. Example: x² − 5x + 6 = 0 gives Δ = 25 − 24 = 1 > 0, yielding roots x = 3 and x = 2.
• Δ = 0: One repeated real root. The parabola is tangent to the x-axis at exactly one point. The root equals x = −b / (2a). Example: x² − 4x + 4 = 0 gives Δ = 0, root x = 2.
• Δ < 0: Two complex conjugate roots of the form (−b ± i√|Δ|) / (2a), where i = √(−1). The parabola does not intersect the x-axis.

Sum and Product of Roots

Vieta's formulas provide two elegant relationships derived directly from the quadratic formula, enabling rapid verification of computed roots:

• Sum of roots: x₁ + x₂ = −b / a
• Product of roots: x₁ × x₂ = c / a

These identities hold even when the roots are complex, making them universally applicable for checking work without substitution.

Worked Example

Consider 2x² − 4x − 6 = 0, where a = 2, b = −4, c = −6.

• Discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64
• First root: x₁ = (4 + √64) / 4 = (4 + 8) / 4 = 3
• Second root: x₂ = (4 − 8) / 4 = −4 / 4 = −1
• Verification — Sum: 3 + (−1) = 2 = −(−4)/2 ✓ | Product: 3 × (−1) = −3 = −6/2 ✓

Real-World Applications

Quadratic equations govern a wide range of real-world phenomena across science, engineering, and finance:

• Projectile motion: An object launched upward follows h(t) = −16t² + v₀t + h₀ (in feet). Setting h = 0 and applying the quadratic formula gives the exact landing time.
• Engineering and architecture: Parabolic arches, satellite dish reflectors, and suspension bridge cable profiles all satisfy quadratic relationships that engineers solve using this formula.
• Finance: Revenue functions of the form R(x) = −ax² + bx + c require the quadratic formula to find break-even quantities where R(x) = 0.
• Physics: Resonance frequency calculations in RLC circuits and lens focal-length equations both reduce to quadratic forms requiring the standard formula.