Question

for quadratic function h, h(-3/2)=0 and h(5)=0. what is a possible equation for h in factored form?
move the correct answer to each box. not all answers will be used.

h(x)= ____, ____.

Options:
(x+5)
(x-5)
(2x-3)
(2x+3)
(3x-2)
(3x+2)

Answer 1

Since the quadratic function \( h \) has roots at \( x = -\frac{3}{2} \) and \( x = 5 \), we can express the function in factored form as follows:

The factors corresponding to the roots are:
1. For the root \( x = -\frac{3}{2} \), the factor is \( \left(x + \frac{3}{2}\right) \).
2. For the root \( x = 5 \), the factor is \( (x - 5) \).

We can multiply by a constant (let's choose 2 for simplicity) to make the factor \( \left(x + \frac{3}{2}\right) \) into an integer form:

1. \( \left(x + \frac{3}{2}\right) = \left(2x + 3\right)/2 \).

Thus the function can be expressed as:

\[
h(x) = k \cdot \left(2x + 3\right)(x - 5)
\]

where \( k \) is a constant (usually we can take \( k = 1 \) for simplicity).

In the provided options, we can use \( (2x + 3) \) and \( (x - 5) \).

So in factor form, possible equations for \( h \) are:

\[
h(x) = (2x + 3), (x - 5).
\] (though this specific answer doesn't exactly match your request for two options).

Given the options presented, the closest representations are:

1. \( (2x + 3) \)
2. \( (x - 5) \).

However, based on the options provided, let's see what we can utilize in the boxes:

- Since \( (x - 5) \) is present as a root factor,
- The term \( (2x + 3) \) matches the necessary transformation we sought for the root \( -\frac{3}{2} \).

Thus the final answer to fill in your boxes will be:

**h(x)= (2x + 3), (x - 5).**