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What is a quadratic polynomial function?

A quadratic polynomial function is a polynomial function of degree 2. It has the general form:

Where:

• a, b, and c are constants
• a ≠ 0 (since it's a quadratic function)
• x is the variable

  The graph of a quadratic function is a parabola. The roots of the function are the x-intercepts of this parabola, where the graph crosses the x-axis.

How to calculate the roots (alpha)

The roots of a quadratic polynomial function can be calculated using the quadratic formula:

This formula gives two roots, which may be real and distinct, real and equal, or complex conjugates depending on the discriminant (b² - 4ac).

Steps to calculate the roots:

1. Identify the coefficients a, b, and c from the quadratic equation
2. Calculate the discriminant: D = b² - 4ac
3. If D > 0, there are two distinct real roots
4. If D = 0, there is one real root (a repeated root)
5. If D < 0, there are two complex conjugate roots
6. Apply the quadratic formula to find the roots

Worked example

Let's find the roots of the quadratic function f(x) = 2x² - 4x - 6.

Step 1: Identify coefficients

a = 2, b = -4, c = -6

Step 2: Calculate discriminant

D = b² - 4ac = (-4)² - 4(2)(-6) = 16 + 48 = 64

Step 3: Determine nature of roots

Since D = 64 > 0, there are two distinct real roots.

Step 4: Apply quadratic formula

α₁ = [-(-4) + √64] / (2*2) = [4 + 8] / 4 = 12/4 = 3

α₂ = [-(-4) - √64] / (2*2) = [4 - 8] / 4 = -4/4 = -1

Final answer

The roots of the function are x = 3 and x = -1.

Interpreting the results

The roots of a quadratic function have several important interpretations:

• They represent the x-intercepts of the parabola
• They indicate where the function crosses the x-axis
• They can be used to factor the quadratic expression
• They help determine the vertex of the parabola

For example, in the worked example above, the function crosses the x-axis at x = 3 and x = -1. This means the parabola has points (3, 0) and (-1, 0) on its graph.