If `A=sqrt((s-a)(s-b)(s-c))`, then express c in terms of A,s,a and b.

Options

A. `c=(A^(2)-a)/((s-a)(s-b))`
B. `c=s-(A^(2))/((s-a)(s-b))`
C. `c=(A^(2)-s)/((s-a)(s-b))`
D. none of these

Explanation

To express \( c \) in terms of \( A, s, a, \) and \( b \) from the equation \( A = \sqrt{(s-a)(s-b)(s-c)} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ A = \sqrt{(s-a)(s-b)(s-c)} \]
Squaring both sides gives us:
\[ A^2 = (s-a)(s-b)(s-c) \]

### Step 2: Expand the right-hand side
Now, we need to expand the right-hand side:
\[ A^2 = (s-a)(s-b)(s-c) \]
First, expand \( (s-a)(s-b) \):
\[ (s-a)(s-b) = s^2 - (a+b)s + ab \]
Now, we multiply this result by \( (s-c) \):
\[ A^2 = (s^2 - (a+b)s + ab)(s-c) \]
Expanding this gives:
\[ A^2 = s^3 - cs^2 - (a+b)s^2 + (a+b)cs + abs - abc \]
Combining like terms, we have:
\[ A^2 = s^3 - (c + a + b)s^2 + (ab + (a+b)c)s - abc \]

### Step 3: Rearranging to isolate \( c \)
To express \( c \) in terms of \( A, s, a, \) and \( b \), we need to isolate \( c \). We can rearrange the equation:
\[ A^2 = (s-a)(s-b)(s-c) \]
to solve for \( (s-c) \):
\[ s-c = \frac{A^2}{(s-a)(s-b)} \]
Now, we can isolate \( c \):
\[ c = s - \frac{A^2}{(s-a)(s-b)} \]

### Final Expression
Thus, the expression for \( c \) in terms of \( A, s, a, \) and \( b \) is:
\[ c = s - \frac{A^2}{(s-a)(s-b)} \]