Question

What could be the factorization of a quadratic with roots x= 0.5 and x = –9? Select ALL that apply. A. f(x)= (x+ 0.5) (x −9) B. f(x)= −(x − 0.5) (x+ 9) C. f(x) = (2x −1)(x+ 9) D. f(x)= (3x+ 6)(3x −27)

Answer 1

To find the factorization of a quadratic given its roots, we can use the fact that if \( r_1 \) and \( r_2 \) are the roots of a quadratic equation, then it can be expressed in the form:

\[
f(x) = k(x - r_1)(x - r_2)
\]

where \( k \) is a non-zero constant.

In this case, we have the roots \( x = 0.5 \) (which is equivalent to \( r_1 = \frac{1}{2} \)) and \( x = -9 \) (which is \( r_2 = -9 \)). Therefore, we can express our quadratic as:

\[
f(x) = k\left(x - 0.5\right)(x + 9)
\]

Now we can expand this and find \( k \):
1. For \( k = 1 \):
\[
f(x) = (x - 0.5)(x + 9)
\]

2. If we set \( k = -1 \):
\[
f(x) = -(x - 0.5)(x + 9)
\]

To express \( f(x) \) without the constant \( k \), we can also create other forms:
- By multiplying the entire quadratic expression by 2:
\[
f(x) = 2\left(x - 0.5\right)(x + 9) = (2x - 1)(x + 9)
\]

- To create another expression, we could multiply the entire expression by 3, for example:
\[
f(x) = 3\left(x - 0.5\right)(x + 9) = (3x - 1.5)(x + 9) \text{ (but this does not exist in the options.)}
\]

Now let’s analyze the given options:

A. \( f(x)= (x+ 0.5) (x −9) \) - This is incorrect. The roots are not 0.5 and -9.

B. \( f(x)= −(x − 0.5) (x+ 9) \) - This is a valid factorization because it corresponds to \( k = -1 \).

C. \( f(x) = (2x −1)(x+ 9) \) - We derived this earlier where it corresponds to \( k = 2 \). This is also valid.

D. \( f(x)= (3x+ 6)(3x −27) \) - This expands to \( 9x^2 - 81 \), which does not match a quadratic with the given roots of 0.5 and -9.

Thus, the correct options are:

**B and C.**