Quadratic Equation Given Roots and Vertex Calculator

This calculator helps you find the quadratic equation in standard form when you know the roots and the vertex of the parabola. The quadratic equation is expressed in the form ax² + bx + c = 0, and we'll determine the coefficients a, b, and c based on the given information.

Introduction

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. The graph of a quadratic equation is a parabola. When you know the roots (x-intercepts) and the vertex of the parabola, you can determine the equation of the quadratic function.

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The standard form can be derived from the vertex form by expanding it.

How to Use the Calculator

Enter the x-coordinate of the first root in the "First Root" field.
Enter the x-coordinate of the second root in the "Second Root" field.
Enter the x-coordinate of the vertex in the "Vertex X" field.
Enter the y-coordinate of the vertex in the "Vertex Y" field.
Click the "Calculate" button to see the quadratic equation in standard form.

The calculator will display the quadratic equation in the form ax² + bx + c = 0, along with the vertex coordinates and the axis of symmetry.

Formula

The quadratic equation can be found using the following steps:

Find the sum and product of the roots: sum = r1 + r2, product = r1 × r2.
Use the vertex form: y = a(x - h)² + k.
Substitute one of the roots into the vertex form to solve for 'a'.
Expand the vertex form to get the standard form: ax² + bx + c = 0.

Vertex Form

y = a(x - h)² + k

Standard Form

ax² + bx + c = 0

Example Calculation

Let's find the quadratic equation with roots at x = 2 and x = 4, and vertex at (3, -1).

Sum of roots: 2 + 4 = 6
Product of roots: 2 × 4 = 8
Using vertex form: y = a(x - 3)² - 1
Substitute x = 2: 0 = a(2 - 3)² - 1 → 0 = a(1) - 1 → a = 1
Vertex form becomes: y = (x - 3)² - 1
Expand to standard form: y = x² - 6x + 9 - 1 → y = x² - 6x + 8

The quadratic equation is x² - 6x + 8 = 0.

FAQ

What is the difference between the vertex form and standard form of a quadratic equation?: The vertex form (y = a(x - h)² + k) shows the vertex of the parabola directly, while the standard form (ax² + bx + c = 0) is more general and can be used for any quadratic equation.
Can I find the quadratic equation if I only know the roots?: Yes, you can use the factored form (y = a(x - r1)(x - r2)) and then expand it to standard form. However, knowing the vertex allows you to determine the coefficient 'a' more directly.
What if the vertex is not given?: If the vertex is not given, you can still find the quadratic equation using the roots, but you won't be able to determine the coefficient 'a' without additional information.
How do I know if the parabola opens upwards or downwards?: The coefficient 'a' determines the direction of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
Can I use this calculator for cubic or higher-degree equations?: No, this calculator is specifically designed for quadratic equations (degree 2). For higher-degree equations, you would need a different calculator.