To factor the quadratic function [tex]\( f(x) = x^2 - 17x + 72 \)[/tex] and write it in intercept form, we follow these steps:
1. Identify the quadratic function:
The given function is [tex]\( f(x) = x^2 - 17x + 72 \)[/tex].
2. Identify the coefficients:
The quadratic function is in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -17 \)[/tex], and [tex]\( c = 72 \)[/tex].
3. Determine two numbers that multiply to [tex]\( c \)[/tex] and add to [tex]\( b \)[/tex]:
We need to find two numbers that multiply to 72 (the constant term [tex]\( c \)[/tex]) and add up to -17 (the coefficient of [tex]\( x \)[/tex]).
These two numbers are -9 and -8 because:
[tex]\[
(-9) \times (-8) = 72 \quad \text{(product)}
\][/tex]
[tex]\[
(-9) + (-8) = -17 \quad \text{(sum)}
\][/tex]
4. Rewrite the quadratic as a product of two binomials:
Using the numbers -9 and -8, we can factor the quadratic expression:
[tex]\[
f(x) = x^2 - 17x + 72 = (x - 9)(x - 8)
\][/tex]
5. Final answer in intercept form:
The function [tex]\( f(x) \)[/tex] in intercept form is:
[tex]\[
f(x) = (x - 9)(x - 8)
\][/tex]
So, the intercept form of the function is:
[tex]\[
f(x) = (x - 9)(x - 8)
\][/tex]
