Quadratic Function From Roots Calculator

A quadratic function is a second-degree polynomial that can be expressed in the form f(x) = ax² + bx + c. When given the roots of the quadratic equation, we can determine the quadratic function that has those roots. This calculator helps you find the quadratic function from its roots quickly and accurately.

What is a Quadratic Function from Roots?

A quadratic function is a polynomial function of degree two. It can be written in the standard form:

f(x) = ax² + bx + c

When we know the roots of the quadratic equation (the values of x that make f(x) = 0), we can determine the quadratic function that has those roots. The roots are the solutions to the equation ax² + bx + c = 0.

Given two roots, r₁ and r₂, the quadratic function can be expressed in its factored form:

f(x) = a(x - r₁)(x - r₂)

Expanding this factored form gives us the standard quadratic form, from which we can identify the coefficients a, b, and c.

The Formula

To find the quadratic function from its roots, follow these steps:

Identify the two roots, r₁ and r₂.
Write the function in factored form: f(x) = a(x - r₁)(x - r₂).
Expand the factored form to get the standard quadratic form.
Identify the coefficients a, b, and c from the expanded form.

The expanded form of the quadratic function is:

f(x) = a(x² - (r₁ + r₂)x + r₁r₂)

From this, we can see that:

a is the leading coefficient.
b = -a(r₁ + r₂)
c = a(r₁r₂)

If the leading coefficient a is not specified, it is typically assumed to be 1.

Worked Example

Let's find the quadratic function with roots at x = 2 and x = -3.

Identify the roots: r₁ = 2, r₂ = -3.
Write the function in factored form: f(x) = a(x - 2)(x + 3).
Expand the factored form:
f(x) = a(x² + 3x - 2x - 6) = a(x² + x - 6)
Identify the coefficients: a = a, b = a, c = -6a.

If we assume a = 1, the quadratic function is:

f(x) = x² + x - 6

This means the quadratic function with roots at x = 2 and x = -3 is f(x) = x² + x - 6.

Interpreting Results

When you use the quadratic function from roots calculator, you'll get the quadratic function in standard form. Here's what each part of the result means:

Leading coefficient (a): Determines the parabola's width and direction. If a is positive, the parabola opens upwards; if negative, it opens downwards.
Linear coefficient (b): Affects the parabola's symmetry and vertex position.
Constant term (c): Shifts the parabola vertically.

The roots of the quadratic function are the x-intercepts of the parabola. The vertex of the parabola can be found using the formula x = -b/(2a).

Note: The quadratic function is unique up to a constant factor. If you don't specify the leading coefficient, the calculator assumes a = 1.

FAQ

What is the difference between the standard form and factored form of a quadratic function?

The standard form is f(x) = ax² + bx + c, while the factored form is f(x) = a(x - r₁)(x - r₂). The factored form is useful when you know the roots, while the standard form is useful for calculations and graphing.

Can I find the quadratic function if I only have one root?

No, you need at least two roots to determine a unique quadratic function. A quadratic equation has exactly two roots (counting multiplicities).

What if the roots are complex numbers?

The calculator can handle complex roots, but the resulting quadratic function will have complex coefficients. The roots are still valid, and the function can be graphed in the complex plane.

How do I find the roots of a quadratic function if I have the standard form?

You can use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). This will give you the two roots of the quadratic function.