Cal11 calculator
Search calculator...

Quadratic Equation That Goes Through The Roots and Vertex Calculator

Reviewed by Calculator Editorial Team

This calculator finds the quadratic equation that passes through two given roots and a specified vertex. It's useful in algebra, physics, and engineering when you need a quadratic function with specific roots and vertex.

How to Use This Calculator

To use this calculator, you'll need to provide:

  1. The first root (x₁) of the quadratic equation
  2. The second root (x₂) of the quadratic equation
  3. The x-coordinate of the vertex (h)
  4. The y-coordinate of the vertex (k)

Enter these values in the calculator panel on the right, then click "Calculate". The calculator will display the quadratic equation in both standard and vertex forms.

The Formula

A quadratic equation that passes through roots x₁ and x₂ can be written in its standard form as:

y = a(x - x₁)(x - x₂)

To include the vertex (h, k), we adjust the equation to:

y = a(x - h)² + k

The calculator combines these two forms to find the quadratic equation that satisfies both conditions.

Worked Example

Let's find the quadratic equation that passes through roots at x = 2 and x = 4, and has a vertex at (1, 3).

  1. Enter x₁ = 2, x₂ = 4, h = 1, and k = 3 in the calculator.
  2. Click "Calculate".
  3. The calculator will display the equation in both forms.

The result shows the quadratic equation that meets all these conditions.

Interpreting Results

The calculator provides two forms of the quadratic equation:

  1. Standard form: y = a(x - x₁)(x - x₂)
  2. Vertex form: y = a(x - h)² + k

The standard form shows the roots clearly, while the vertex form highlights the vertex coordinates. The coefficient 'a' determines the parabola's width and direction.

FAQ

What if I don't know the vertex coordinates?
You can use our Vertex of a Parabola Calculator to find the vertex first, then use those values here.
Can the quadratic equation have a negative coefficient 'a'?
Yes, the calculator will determine the correct value of 'a' based on your inputs to ensure the equation passes through the specified points.
What if the roots are the same?
The calculator will still work, but the resulting equation will have a double root at that x-value.
How accurate are the results?
The calculator uses precise mathematical calculations to ensure accurate results based on your inputs.