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Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula. Find real and complex roots, discriminant, vertex, and step-by-step solution.

Quadratic Formula Worksheet
ax² + bx + c = 0

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The Quadratic Formula

For any equation ax² + bx + c = 0 where a ≠ 0, the solutions are given by:

x = (-b ± √(b² - 4ac)) / 2a

Discriminant (b² - 4ac)

The discriminant determines the nature of the roots:

  • Positive: Two distinct real roots
  • Zero: One repeated real root (parabola touches x-axis)
  • Negative: Two complex conjugate roots (parabola doesn't cross x-axis)

Examples

  • x² - 5x + 6 = 0 → x = 2 or x = 3 (discriminant = 1)
  • x² - 4x + 4 = 0 → x = 2 (discriminant = 0)
  • x² + x + 1 = 0 → x = -0.5 ± 0.866i (discriminant = -3)

Vertex Form

The vertex of the parabola y = ax² + bx + c is at (-b/2a, c - b²/4a). The axis of symmetry is x = -b/2a.

Other Methods for Solving Quadratics

Factoring

If the equation factors neatly, this is the fastest method. x² - 5x + 6 = 0 factors to (x-2)(x-3) = 0, giving x = 2 or x = 3. Not all quadratics factor into integers — the quadratic formula works for every case.

Completing the Square

Rearrange ax² + bx + c = 0 into vertex form a(x-h)² + k = 0. This method is how the quadratic formula itself is derived. It's useful when you need the vertex for graphing.

Real-World Applications

  • Projectile motion: The height of a thrown ball follows h(t) = -16t² + v₀t + h₀ (in feet). Setting h = 0 and solving gives the time when it hits the ground
  • Area problems: If a rectangle has perimeter 30 and area 50, the dimensions satisfy x² - 15x + 50 = 0, giving x = 5 or x = 10
  • Revenue optimization: Revenue R = price × quantity, where quantity often depends linearly on price, creating a quadratic equation. The vertex gives the price that maximizes revenue
  • Engineering: Structural load calculations, circuit analysis, and signal processing all use quadratic equations

Quick Tips

  • If c = 0, the equation simplifies: ax² + bx = 0 → x(ax + b) = 0 → x = 0 or x = -b/a
  • If b = 0, solve directly: ax² + c = 0 → x = ±√(-c/a)
  • The sum of the roots equals -b/a and the product equals c/a (Vieta's formulas)

Graphing Quadratic Functions

The graph of y = ax² + bx + c is always a parabola. Key features you can determine algebraically:

  • Direction: Opens upward if a > 0, downward if a < 0
  • Vertex: The turning point at x = -b/(2a). This is the minimum (a > 0) or maximum (a < 0) value
  • Axis of symmetry: The vertical line x = -b/(2a). The parabola is symmetric about this line
  • Y-intercept: The constant term c (set x = 0)
  • X-intercepts: The roots found by the quadratic formula (if real)

History of the Quadratic Formula

Babylonian mathematicians solved quadratic equations as early as 2000 BC, though they used geometric methods rather than a formula. The Indian mathematician Brahmagupta (628 AD) provided one of the earliest explicit solutions for the general quadratic equation. The formula as we know it today was formalized by European mathematicians during the Renaissance. Despite being over 1,000 years old, the quadratic formula remains one of the most frequently used tools in algebra, engineering, physics, and computer graphics.

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Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, and it solves any equation of the form ax² + bx + c = 0 where a is not zero. If a equals zero, the equation becomes bx + c = 0, which is linear. The formula only works for degree-2 equations, not cubic or higher ones.

What does the discriminant tell you?

The discriminant, b² - 4ac, tells you how many real roots a quadratic has. If it is positive there are two different real roots. If it is zero there is one repeated root at x = -b / 2a, where the parabola just touches the x-axis at its vertex. If it is negative there are no real roots.

What happens when the discriminant is negative?

A negative discriminant gives two complex conjugate roots and no real ones. The square root of a negative number is imaginary, so the roots take the form p + qi and p - qi, where i = √(-1). On a graph, the parabola never crosses or touches the x-axis.

How do you solve x² - 5x + 6 = 0?

The roots are x = 2 and x = 3. With a = 1, b = -5 and c = 6, the discriminant is 25 - 24 = 1, so x = (5 ± 1) / 2. Check by substitution: 4 - 10 + 6 = 0. The roots also match the shortcuts sum = -b/a = 5 and product = c/a = 6.