Quadratic Function Calculator Given Roots

When you know the roots of a quadratic equation, you can easily determine the quadratic function that produced those roots. This calculator helps you find the quadratic equation in its standard form when you provide the two roots.

Introduction

A quadratic function is a second-degree polynomial that can be written in the standard form:

f(x) = ax² + bx + c

When you know the roots (x₁ and x₂) of the quadratic equation, you can determine the coefficients a, b, and c. The roots are the values of x that satisfy the equation f(x) = 0.

How to Use the Calculator

Enter the first root (x₁) in the first input field.
Enter the second root (x₂) in the second input field.
Optionally, you can specify the leading coefficient (a) if you want a specific value for it. If left blank, the calculator will assume a = 1.
Click the "Calculate" button to see the quadratic function.
The result will be displayed in the standard form f(x) = ax² + bx + c.

Formula

Given two roots x₁ and x₂, the quadratic function can be expressed as:

f(x) = a(x - x₁)(x - x₂)

Expanding this expression gives the standard form:

f(x) = ax² - a(x₁ + x₂)x + a x₁ x₂

From this, we can identify the coefficients:

a = a (the leading coefficient you specify or default to 1)
b = -a(x₁ + x₂)
c = a x₁ x₂

Worked Example

Let's find the quadratic function with roots at x₁ = 2 and x₂ = -3, and a leading coefficient of a = 2.

Using the formula:

f(x) = 2(x - 2)(x - (-3)) = 2(x - 2)(x + 3)

Expanding the expression:

f(x) = 2(x² + 3x - 2x - 6) = 2(x² + x - 6) = 2x² + 2x - 12

So, the quadratic function is f(x) = 2x² + 2x - 12.

Interpreting Results

The result of the calculation is the quadratic function in its standard form. This equation represents a parabola that passes through the given roots and has the specified leading coefficient.

You can use this function to:

Graph the quadratic function
Find the vertex of the parabola
Determine the axis of symmetry
Find the maximum or minimum value of the function

FAQ

What if I only know one root?

If you only know one root, you cannot uniquely determine the quadratic function. A quadratic equation with a single known root could have infinitely many solutions depending on the leading coefficient and the other root.

Can the roots be complex numbers?

Yes, the roots can be complex numbers. The calculator will handle complex roots correctly and provide the corresponding quadratic function.

What if both roots are the same?

If both roots are the same (a repeated root), the quadratic function will have a double root. The calculator will still work and provide the correct quadratic function.

Can I use this calculator for cubic or higher-degree polynomials?

No, this calculator is specifically designed for quadratic functions (degree 2). For higher-degree polynomials, you would need a different calculator or method.