\text(g(x) = 2x^2+18x-14) : Answers A. g(9) B. g(3x) C. g(1+5m)

Let's evaluate the given function \( g(x) = 2x^2 + 18x - 14 \) for each of the requested inputs.

### A. \( g(9) \)

We substitute \( x = 9 \) into the function:

\[
g(9) = 2(9)^2 + 18(9) - 14
\]
\[
= 2(81) + 162 - 14
\]
\[
= 162 + 162 - 14 = 324 - 14 = 310
\]

Thus, \( g(9) = 310 \).

### B. \( g(3x) \)

We substitute \( x = 3x \) into the function:

\[
g(3x) = 2(3x)^2 + 18(3x) - 14
\]
\[
= 2(9x^2) + 54x - 14
\]
\[
= 18x^2 + 54x - 14
\]

Thus, \( g(3x) = 18x^2 + 54x - 14 \).

### C. \( g(1 + 5m) \)

We substitute \( x = 1 + 5m \) into the function:

\[
g(1 + 5m) = 2(1 + 5m)^2 + 18(1 + 5m) - 14
\]
First, expand \( (1 + 5m)^2 \):

\[
(1 + 5m)^2 = 1^2 + 2(1)(5m) + (5m)^2 = 1 + 10m + 25m^2
\]

Now substitute this back into the equation:

\[
g(1 + 5m) = 2(1 + 10m + 25m^2) + 18(1 + 5m) - 14
\]
\[
= 2(1) + 2(10m) + 2(25m^2) + 18(1) + 18(5m) - 14
\]
\[
= 2 + 20m + 50m^2 + 18 + 90m - 14
\]
\[
= 50m^2 + (20m + 90m) + (2 + 18 - 14)
\]
\[
= 50m^2 + 110m + 6
\]

Thus, \( g(1 + 5m) = 50m^2 + 110m + 6 \).

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Do you need more details or have any questions?

### Related Questions:
1. How can you find the derivative of \( g(x) \)?
2. What is the vertex of the parabola defined by \( g(x) \)?
3. How does changing the input to \( 3x \) affect the graph of the function?
4. What happens to \( g(x) \) if you evaluate it for negative values of \( x \)?
5. Can you factorize \( g(x) = 2x^2 + 18x - 14 \)?

### Tip:
For any function \( g(x) \), plugging in different expressions for \( x \) modifies the result according to the new input's structure, which can also change the graph of the function.

g(x) = 2x^2 + 18x - 14 : Answers A. g(9) B. g(3x) C. g(1+5m)

This solution explains how to evaluate the quadratic function g(x) = 2x^2 + 18x - 14 for different inputs, including g(9), g(3x), and g(1 + 5m). It covers algebraic substitution and quadratic expansion concepts, suitable for Grades 9-10 students.

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