How to write a quadratic function in Factored Form
In this article we explain how to write a quadratic function in its factored form, that is, using zeros to get an equation in product form.
Table of Contents
What is the factored form?
Given a quadratic function *f(x)=ax^2+bx+c*, we can write it as a product using its zeros and leading coefficient. This is known as the factored form of the quadratic function:
*f(x)=a(x-x_1)(x-x_2)*
*x_1* and *x_2* are the zeros of the function.
If the discriminant is zero, the zeros will be equal *x_1=x_2*, and the equation takes the form:
*f(x)=a(x-x_1)^2*
Since we are dealing with real functions, we will not consider the case where the discriminant is negative, as the zeros would be complex numbers.
It is evident that to find the factored form, we must first know the zeros of the function. For this, it is necessary to review this article:
Solved exercises
Exercise 1: Find the factored form of the function *f(x)=3x^2+3x-18*
Solution:
Using the quadratic formula, we find the zeros of *f:* *x_1=2, x_2=-3.* Write the function in factored form as:
*f(x)=a(x-x_1)(x-x_2)*
*f(x)=3(x-2)(x-(-3))*
*f(x)=3(x-2)(x+3)*
Exercise 2: Write the function *f(x)=-5x^2-10x-5* in factored form.
Solution:
Find the zeros of the function, noticing that there is only one zero with multiplicity two: *x_1=-1.* Write the factored form as:
*f(x)=a(x-x_1)(x-x_2)*
As *x_1=x_2,* it results in:
*f(x)=a(x-x_1)^2*
*f(x)=-5(x-(-1))^2*
*f(x)=-5(x+1)^2*
Exercise 3: Factorize the quadratic function *f(x)=x^2-16*
Solution:
Calculate the zeros: *x_1=-4, x_2=4* and write the factored form as:
*f(x)=a(x-x_1)(x-x_2)*
*f(x)=1(x-(-4))(x-4)*
*f(x)=(x+4)(x-4)*
Exercise 4: Find the factored form of the function *f(x)=-7x^2*
Solution:
Find the zeros and notice there is only one zero with multiplicity two: *x_1=0,* in this case, it corresponds to the factored form:
*f(x)=a(x-x_1)^2*
*f(x)=-7(x-0)^2*
*f(x)=-7x^2*
We see that in this case, the factored form and the polynomial form coincide.
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