Quadratic Formula Calculator — Solve Ax² + Bx + C = 0

What is the Quadratic Formula?

The quadratic formula is the solution to a second-degree polynomial equation of the form:

Ax² + Bx + C = 0

If you can rewrite your equation in this form, it can be solved with the quadratic formula. The formula is:

x = (−B ± √Δ) / 2A

where the discriminant Δ = B² − 4AC determines the nature of the roots.

Coefficients of a Quadratic Equation

A, B, and C are the coefficients of the quadratic equation. They are all real numbers, not dependent on x.

• A — the leading coefficient (coefficient of x²). Must not be zero, otherwise the equation is linear, not quadratic.
• B — the middle coefficient (coefficient of x). Can be any real number including zero.
• C — the constant term. Can be any real number including zero.

If B² < 4AC, then the discriminant Δ will be negative, meaning the equation has no real roots.

Three Cases for the Discriminant

• Δ > 0 — Two unique real roots:
  x₁ = (−B + √Δ) / 2A    and    x₂ = (−B − √Δ) / 2A
• Δ = 0 — One repeated (double) root:
  x = −B / 2A
• Δ < 0 — No real solutions; the roots are complex conjugates:
  x₁ = −B/2A + i·√(−Δ)/2A    and    x₂ = −B/2A − i·√(−Δ)/2A

How to Use the Quadratic Formula Solver

1. Write down your equation. For example: 4x² + 3x − 7 = −4 − x
2. Bring to standard form Ax² + Bx + C = 0:
   4x² + 3x − 7 = −4 − x
   4x² + (3 + 1)x + (−7 + 4) = 0
   4x² + 4x − 3 = 0   →   A = 4, B = 4, C = −3
3. Calculate the discriminant:
   Δ = B² − 4AC = 4² − 4 × 4 × (−3) = 16 + 48 = 64
4. Since Δ = 64 > 0, there are two real roots:
   x₁ = (−4 + √64) / (2 × 4) = (−4 + 8) / 8 = 4/8 = 0.5
   x₂ = (−4 − √64) / (2 × 4) = (−4 − 8) / 8 = −12/8 = −1.5

Solving Quadratic Equations with a Negative Discriminant

When Δ < 0, the equation has no real solutions — the parabola y = Ax² + Bx + C does not cross the x-axis. The roots are complex numbers:

x = −B/(2A) ± i · √(−Δ)/(2A)

For example, x² + 4 = 0 gives Δ = 0 − 16 = −16, so x₁ = 2i and x₂ = −2i.

Application: Golden Ratio

The golden ratio φ ≈ 1.618 is the positive root of the quadratic equation:

x² − x − 1 = 0

Here A = 1, B = −1, C = −1. The discriminant is Δ = 1 + 4 = 5, giving:

• φ = (1 + √5) / 2 ≈ 1.6180339...
• ψ = (1 − √5) / 2 ≈ −0.6180339...

The golden ratio appears in architecture, art, nature (sunflower spirals, shell growth), and many areas of mathematics. Click the φ (x²−x−1=0) quick example button above to verify it instantly.

The Parabola Vertex

The quadratic function y = Ax² + Bx + C traces a parabola. Its vertex (the highest or lowest point) is located at:

• x_v = −B / (2A)
• y_v = C − B² / (4A)

The roots of the equation are the x-intercepts of this parabola.

Real-World Applications

• Physics — projectile motion, free fall (when does an object hit the ground?)
• Engineering — parabolic arches, vertical curve design for roads
• Finance — break-even analysis, profit maximization (when does revenue equal cost?)
• Construction — calculating dimensions of curved surfaces and spans
• Optics — parabolic mirror focus distance

Unit Support

This calculator supports an optional unit label for the variable x, covering:

• US Imperial: inches, feet, yards, miles, ounces, pounds
• Metric: mm, cm, m, km, g, kg, sq meters, sq cm
• Time: seconds, minutes, hours
• Currencies: US Dollar ($), Russian Ruble (₽), Euro (€), British Pound (£), Japanese Yen (¥), Chinese Yuan, Canadian Dollar, Australian Dollar, Swiss Franc, Indian Rupee, Brazilian Real, Korean Won, Swedish Krona