Final answer:
The value of 'a' in the given functions f(x)=5x^(2)-3 and f(x+a)=5x^(2)+30x+42 is 3, as it satisfies both equations.
Step-by-step explanation:
The value of 'a' can only be determined when the right sides of the two functions match. From these functions, we can write f(x+a) in terms of f(x) and then compare the two functions. The function f(x) is 5x^(2)-3, and f(x+a) should be 5(x+a)^(2)-3.
Expanding this gives 5x^(2)+10ax+5a^(2)-3. But we were given f(x+a) as 5x^(2)+30x+42, so these two forms of f(x+a) should be equal. This leads to the equations 10ax=30x and 5a^(2)-3=42.
From the first equation, we get a=3. Substituting 'a=3' into the second equation, we get 45-3=42, which holds true. Therefore, the value of 'a' that satisfies both functions is 3.
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