Question

Determine g(x + a) – g(x) for the following function.
g(x) = 5x2 + 2x

Answer 1

`g(x + a) - g(x) = 10ax + 5a^2 + 2a`

Explanation:

We are given the function:

`g(x) = 5x^2 + 2x`

And we want to determine:

`\displaystyle g(x+a) - g(x)`

Substitute:

`\displaystyle \begin{aligned}g(x + a) - g(x) &=\left[5(x+a)^2 + 2(x+a)\right] -\left[5x^2+2x\right] \end{aligned}`

And simplify:

`\displaystyle \begin{aligned}g(x + a) - g(x) &=\left[5(x+a)^2 + 2(x+a)\right] -\left[5x^2+2x\right] \\ \\ &= \left(5(x^2 + 2ax + a^2) + (2x + 2a) \right) + \left(-5x^2 - 2x\right) \\ \\ &= \left((5x^2 + 10ax + 5a^2) + (2x + 2a)\right) + \left(-5x^2 - 2x\right) \\ \\ &= (5x^2-5x^2) + (10ax + 2x - 2x) + (5a^2+2a) \\ \\ &= 10ax + 5a^2 + 2a \end{aligned}`

In conclusion:

`g(x + a) - g(x) = 10ax + 5a^2 + 2a`