If f(x) = 5x² and g(x) = x² + 12x + 85, what is the sum of all values for a such that f(a + 2) = g(2a)?

To solve the problem where f ( a + 2 ) = g ( 2 a ) given f ( x ) = 5 x 2 and g ( x ) = x 2 + 12 x + 85 , we need to follow these steps:

Substitute the expressions given:


Substitute a + 2 into f ( x ) :
f ( a + 2 ) = 5 ( a + 2 ) 2 = 5 ( a 2 + 4 a + 4 ) = 5 a 2 + 20 a + 20

Substitute 2 a into g ( x ) :
g ( 2 a ) = ( 2 a ) 2 + 12 ( 2 a ) + 85 = 4 a 2 + 24 a + 85



Set the two expressions equal to each other:
[


5a^2 + 20a + 20 = 4a^2 + 24a + 85 ]

Move all terms to one side of the equation to set it to zero:
[


5a^2 + 20a + 20 - 4a^2 - 24a - 85 = 0 ]
Simplify the equation:
a 2 − 4 a − 65 = 0

Factor the quadratic equation:
We need factors of − 65 that add up to − 4 . The factors of − 65 are − 13 and 5 :
[


(a - 13)(a + 5) = 0 ]

Solve for a :


If a − 13 = 0 , then a = 13 .

If a + 5 = 0 , then a = − 5 .



Find the sum of all values for a :
The values found are a = 13 and a = − 5 . Therefore, their sum is:
13 + ( − 5 ) = 8


Therefore, the sum of all values for a such that f ( a + 2 ) = g ( 2 a ) is 8 .

Answered by AvaCharlotteMiller | 2025-07-22

The sum of all values for a such that f ( a + 2 ) = g ( 2 a ) is 8 . This was found by setting the functions equal, simplifying, and solving the resulting quadratic equation. The solutions to the equation were a = 13 and a = − 5 .
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Answered by AvaCharlotteMiller | 2025-07-30