To find the quadratic equation given the zeros of the function, follow these steps:
1. Identify the Zeros:
The zeros (roots) of the quadratic function are given as [tex]\(x = 1\)[/tex] and [tex]\(x = 4\)[/tex].
2. Form Factors from the Zeros:
Using the zeros, we can form the factors of the quadratic equation. If a quadratic equation has roots [tex]\(x = a\)[/tex] and [tex]\(x = b\)[/tex], then the equation can be expressed in its factored form as [tex]\((x - a)(x - b)\)[/tex].
For the given roots:
- For [tex]\(x = 1\)[/tex], the factor is [tex]\((x - 1)\)[/tex].
- For [tex]\(x = 4\)[/tex], the factor is [tex]\((x - 4)\)[/tex].
3. Multiply the Factors:
Multiply these two binomials to get the quadratic equation:
[tex]\[
(x - 1)(x - 4)
\][/tex]
4. Expand the Product:
Expand the expression by using the distributive property (also known as FOIL method for binomials):
[tex]\[
(x - 1)(x - 4) = x(x - 4) - 1(x - 4)
\][/tex]
Further breaking it down:
[tex]\[
x(x - 4) = x^2 - 4x
\][/tex]
[tex]\[
-1(x - 4) = -x + 4
\][/tex]
5. Combine Like Terms:
Combine all the terms to get the final quadratic equation:
[tex]\[
x^2 - 4x - x + 4 = x^2 - 5x + 4
\][/tex]
Thus, the quadratic equation with zeros 1 and 4 is:
[tex]\[
x^2 - 5x + 4
\][/tex]
Therefore, the quadratic function with the given zeros can be written as [tex]\(y = x^2 - 5x + 4\)[/tex].
