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How to write the factored form of a quadratic function?

Published in Quadratic Functions 3 mins read

To write the factored form of a quadratic function, you express it as a product of linear factors. Let's explore this process with examples.

Understanding Factored Form

The factored form of a quadratic function is generally expressed as:

f(x) = a(x - r1)(x - r2)

Where:

  • a is a constant.
  • r1 and r2 are the roots or zeros of the quadratic function. These are the x-values where the function equals zero.

Steps to Writing the Factored Form

  1. Find the Zeros (Roots): Determine the values of x for which f(x) = 0. This can be done by:

    • Factoring the quadratic expression.
    • Using the quadratic formula.
    • Graphing the function and finding the x-intercepts.

  2. Write the Factors: If r1 and r2 are the zeros, then (x - r1) and (x - r2) are the factors.

  3. Include the Leading Coefficient: Make sure to include the leading coefficient a in front of the factored form. If the original quadratic is in the form ax2 + bx + c, then 'a' is the coefficient of the x2 term. If the quadratic is simply x2 + bx + c, then a = 1.

Examples

Let's look at a couple of examples, including one derived from the provided reference.

Example 1: Difference of Squares (From Reference)

Suppose we have the function: f(x) = x2 - 36. This is a difference of squares.

  • We can factor this directly: f(x) = (x + 6)(x - 6)

  • In this case, r1 = -6 and r2 = 6. According to the reference, "we can factor this to f of x equals x plus six times X minus six now that we have the factored form of our function."

Example 2: Simple Quadratic

Let’s say you have the quadratic function f(x) = x2 - 5x + 6

  1. Find the zeros: We need to find two numbers that multiply to 6 and add up to -5. Those numbers are -2 and -3. Thus, the roots are x = 2 and x = 3.

  2. Write the factors: The factors are (x - 2) and (x - 3).

  3. Include the leading coefficient: In this case, the leading coefficient 'a' is 1.

So, the factored form is: f(x) = (x - 2)(x - 3)

Importance of Factored Form

The factored form is useful because:

  • Finding Zeros: It immediately reveals the zeros of the function. By setting each factor to zero, you can easily solve for x.

  • Graphing: It helps in sketching the graph of the quadratic function by knowing the x-intercepts (zeros).

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