Sure, let's make \( s \) the subject of the formula in the given equation:
[tex]\[
x = \sqrt{\frac{s(s - a)}{b c}}
\][/tex]
We will solve this step-by-step.
### Step 1: Remove the square root by squaring both sides
First, square both sides of the equation to eliminate the square root on the right-hand side.
[tex]\[
x^2 = \left(\sqrt{\frac{s(s - a)}{b c}}\right)^2
\][/tex]
[tex]\[
x^2 = \frac{s(s - a)}{b c}
\][/tex]
### Step 2: Isolate the term involving \( s \)
Next, multiply both sides of the equation by \( b c \) to isolate the term involving \( s \) on one side.
[tex]\[
x^2 \cdot b c = s(s - a)
\][/tex]
[tex]\[
b c x^2 = s^2 - a s
\][/tex]
### Step 3: Bring all terms to one side
Rearrange the equation to bring all terms to one side, setting it up as a quadratic equation in \( s \).
[tex]\[
s^2 - a s - b c x^2 = 0
\][/tex]
### Step 4: Solve the quadratic equation
To solve the quadratic equation \( s^2 - a s - b c x^2 = 0 \), we use the quadratic formula:
[tex]\[
s = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\][/tex]
For our equation \( s^2 - a s - b c x^2 = 0 \), the coefficients are:
\( A = 1 \), \( B = -a \), and \( C = -b c x^2 \).
Substitute these values into the quadratic formula:
[tex]\[
s = \frac{-(-a) \pm \sqrt{(-a)^2 - 4(1)(-b c x^2)}}{2(1)}
\][/tex]
[tex]\[
s = \frac{a \pm \sqrt{a^2 + 4 b c x^2}}{2}
\][/tex]
### Step 5: Simplify the solutions
Now, simplify the expression to get the final solutions for \( s \):
[tex]\[
s = \frac{a + \sqrt{a^2 + 4 b c x^2}}{2} \quad \text{or} \quad s = \frac{a - \sqrt{a^2 + 4 b c x^2}}{2}
\][/tex]
### Conclusion
Therefore, the solutions for \( s \) in terms of \( x \) are:
[tex]\[
s = \frac{a + \sqrt{a^2 + 4 b c x^2}}{2} \quad \text{and} \quad s = \frac{a - \sqrt{a^2 + 4 b c x^2}}{2}
\][/tex]
So, we have found two possible values for \( s \) as functions of \( a \), \( b \), \( c \), and \( x \):
[tex]\[
s = \frac{a + \sqrt{a^2 + 4 b c x^2}}{2}
\][/tex]
[tex]\[
s = \frac{a - \sqrt{a^2 + 4 b c x^2}}{2}
\][/tex]
