How would I rearrange the following algebraic fraction into the required form?

[sktsasus]

Let \(\displaystyle \,f(x)\, =\, -7\, -\, 3\, \sqrt{\strut x\,}.\,\) Then the expression

. . . . .\(\displaystyle \dfrac{f(x\, +\, h)\, -\, f(x)}{h}\)

can be written in the form

. . . . .\(\displaystyle \dfrac{A}{\left(\sqrt{\strut Bx\, +\, Ch\,}\,\right)\, +\, \left(\sqrt{\strut x\,}\,\right)},\)

where A, B, and C are constants. Find the values of A, B, and C.

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I have managed to change f(x) into the one required with all the h's and have simplified it by rationalizing the numerator. This is the fraction I got:

. . . . .\(\displaystyle \dfrac{3\, \left(\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, +\, h\,}\right)}{h}\)

But I am not sure how to change it into the required form asked in the question above and am therefore unable to find the values of A, B and C.

So if anybody could be so kind to say whether or not my fraction above is correct, and, if so, how I should proceed, I would be deeply grateful.

Thank you!

 

[Deleted member 4993]

Let \(\displaystyle \,f(x)\, =\, -7\, -\, 3\, \sqrt{\strut x\,}.\,\) Then the expression

. . . . .\(\displaystyle \dfrac{f(x\, +\, h)\, -\, f(x)}{h}\)

can be written in the form

. . . . .\(\displaystyle \dfrac{A}{\left(\sqrt{\strut Bx\, +\, Ch\,}\,\right)\, +\, \left(\sqrt{\strut x\,}\,\right)},\)

where A, B, and C are constants. Find the values of A, B, and C.

---

I have managed to change f(x) into the one required with all the h's and have simplified it by rationalizing the numerator. This is the fraction I got:

. . . . .\(\displaystyle \dfrac{3\, \left(\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, +\, h\,}\right)}{h}\)

But I am not sure how to change it into the required form asked in the question above and am therefore unable to find the values of A, B and C.

So if anybody could be so kind to say whether or not my fraction above is correct, and, if so, how I should proceed, I would be deeply grateful.

Thank you!

Multiply the numerator and the denominator by [(√x + √(x+h)] and use the fact that a² - b² = (a+b)(a-b).