If `f(x)=5x^2 -3` and `f(x+a)=5x^2 + 30x + 42`, what is the value of a ?

Options

A. `-30`
B. `-3`
C. 3
D. 30

Explanation

To solve the problem, we need to find the value of \( a \) given the functions \( f(x) = 5x^2 - 3 \) and \( f(x + a) = 5x^2 + 30x + 42 \).

### Step-by-step Solution:

1. **Write down the function \( f(x + a) \)**:
We know that \( f(x) = 5x^2 - 3 \). To find \( f(x + a) \), we substitute \( x + a \) into the function:
\[ f(x + a) = 5(x + a)^2 - 3 \]

2. **Expand \( (x + a)^2 \)**:
\[ (x + a)^2 = x^2 + 2ax + a^2 \]
Therefore,
\[ f(x + a) = 5(x^2 + 2ax + a^2) - 3 \]

3. **Distribute the 5**:
\[ f(x + a) = 5x^2 + 10ax + 5a^2 - 3 \]

4. **Combine like terms**:
\[ f(x + a) = 5x^2 + 10ax + (5a^2 - 3) \]

5. **Set the two expressions for \( f(x + a) \) equal**:
We are given that \( f(x + a) = 5x^2 + 30x + 42 \). Therefore, we can set the two expressions equal to each other:
\[ 5x^2 + 10ax + (5a^2 - 3) = 5x^2 + 30x + 42 \]

6. **Compare coefficients**:
- Coefficient of \( x^2 \): \( 5 = 5 \) (This is true.)
- Coefficient of \( x \): \( 10a = 30 \)
- Constant term: \( 5a^2 - 3 = 42 \)

7. **Solve for \( a \)**:
From the coefficient of \( x \):
\[ 10a = 30 \implies a = \frac{30}{10} = 3 \]

8. **Verify with the constant term**:
Substitute \( a = 3 \) into the constant term equation:
\[ 5(3^2) - 3 = 5(9) - 3 = 45 - 3 = 42 \]
This is correct.

Thus, the value of \( a \) is \( \boxed{3} \).