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Root and Vertex Calculator

Reviewed by Calculator Editorial Team

Quadratic equations are fundamental in algebra and have wide applications in physics, engineering, and economics. This calculator helps you find the roots and vertex of any quadratic equation in the standard form.

What is Root and Vertex?

A quadratic equation is a second-degree polynomial equation in the form:

ax² + bx + c = 0

The roots (or solutions) of the equation are the values of x that satisfy the equation. The vertex of a parabola represented by the quadratic equation is the point where the parabola changes direction, which is either a minimum or maximum point.

For a quadratic equation in standard form, the roots can be found using the quadratic formula, and the vertex can be determined using vertex formulas.

How to Use This Calculator

  1. Enter the coefficients a, b, and c of your quadratic equation in the input fields.
  2. Click the "Calculate" button to find the roots and vertex.
  3. View the results including the roots and vertex coordinates.
  4. Use the reset button to clear the inputs and start over.

The calculator will display the roots and vertex of the quadratic equation based on the coefficients you provide.

Quadratic Equation Formula

The roots of a quadratic equation can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are coefficients of the quadratic equation
  • √(b² - 4ac) is the discriminant

The discriminant determines the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root.
  • If the discriminant is negative, there are two complex roots.

Vertex Formula

The vertex of a parabola represented by a quadratic equation can be found using the following formulas:

x-coordinate of vertex = -b / (2a) y-coordinate of vertex = c - (b² / (4a))

The vertex represents the minimum or maximum point of the parabola, depending on the sign of the coefficient a.

Example Calculation

Let's find the roots and vertex of the quadratic equation x² - 5x + 6 = 0.

  1. Identify the coefficients: a = 1, b = -5, c = 6.
  2. Calculate the discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Find the roots using the quadratic formula:
    x = [5 ± √1] / 2 x₁ = (5 + 1)/2 = 3 x₂ = (5 - 1)/2 = 2
  4. Find the vertex coordinates:
    x-coordinate = -(-5)/(2*1) = 2.5 y-coordinate = 6 - ((-5)²)/(4*1) = 6 - 6.25 = -0.25

The roots are x = 2 and x = 3, and the vertex is at (2.5, -0.25).

FAQ

What is the difference between roots and vertex?
Roots are the solutions to the quadratic equation, while the vertex is the minimum or maximum point of the parabola represented by the equation.
How do I know if my quadratic equation has real roots?
Check the discriminant (b² - 4ac). If it's positive, there are two distinct real roots. If it's zero, there's exactly one real root. If it's negative, there are no real roots.
Can I use this calculator for any quadratic equation?
Yes, this calculator works for any quadratic equation in the standard form ax² + bx + c = 0, where a ≠ 0.
What if the coefficient a is zero?
If a is zero, the equation is no longer quadratic and cannot be solved using this calculator. You would need to use linear equation methods instead.