Given that the function f(x)=(a^x)+b passes through the points (c,5) and (2c,161), where b and c are positive constants, what is the value of b?

Given that the function \( f(x) = a^x + b \) passes through the points \( (c, 5) \) and \( (2c, 161) \), we need to find the value of \( b \).

### Step 1: Set up the equations using the given points.

For the point \( (c, 5) \):
\[
f(c) = a^c + b = 5 \quad \text{(Equation 1)}
\]

For the point \( (2c, 161) \):
\[
f(2c) = a^{2c} + b = 161 \quad \text{(Equation 2)}
\]

### Step 2: Subtract Equation 1 from Equation 2.

Subtracting Equation 1 from Equation 2 eliminates \( b \):
\[
a^{2c} + b - (a^c + b) = 161 - 5
\]
\[
a^{2c} - a^c = 156
\]

### Step 3: Factor the equation.

Notice that the equation can be factored as:
\[
a^c(a^c - 1) = 156
\]

### Step 4: Express \( a^c \) from Equation 1.

From Equation 1:
\[
a^c + b = 5 \implies a^c = 5 - b
\]

### Step 5: Substitute \( a^c = 5 - b \) into the factored equation.

Substituting \( a^c = 5 - b \) into \( a^c(a^c - 1) = 156 \):
\[
(5 - b)((5 - b) - 1) = 156
\]
\[
(5 - b)(4 - b) = 156
\]

### Step 6: Expand and solve the quadratic equation.

Expanding the equation:
\[
20 - 9b + b^2 = 156
\]
\[
b^2 - 9b + 20 - 156 = 0
\]
\[
b^2 - 9b - 136 = 0
\]

### Step 7: Solve the quadratic equation.

To solve for \( b \), use the quadratic formula:
\[
b = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-136)}}{2(1)}
\]
\[
b = \frac{9 \pm \sqrt{81 + 544}}{2}
\]
\[
b = \frac{9 \pm \sqrt{625}}{2}
\]
\[
b = \frac{9 \pm 25}{2}
\]

### Step 8: Determine the correct value of \( b \).

The two possible solutions for \( b \) are:
\[
b = \frac{9 + 25}{2} = 17 \quad \text{or} \quad b = \frac{9 - 25}{2} = -8
\]

Since \( b \) is a positive constant, we choose \( b = 17 \).

Thus, the value of \( b \) is \( \boxed{17} \).

Would you like more details on any step, or do you have any other questions?

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Here are some related questions you might find interesting:
1. How would you find the value of \( a \) given \( b = 17 \)?
2. What would be the next step if we wanted to find the value of \( c \)?
3. Can the quadratic equation be solved using any other method?
4. What if \( c \) were negative—how would that affect the solution?
5. How can you verify the value of \( b \) using the original equations?
6. What are some real-world applications of exponential functions like \( f(x) \)?
7. How would the solution change if the points were \( (c, 5) \) and \( (3c, 161) \)?
8. What if the function were \( f(x) = a^x - b \) instead? How would you approach it?

**Tip:** When dealing with quadratic equations, always check for possible extraneous solutions, especially when constraints are given (like \( b \) being positive).

Learn how to find the value of b in the exponential function f(x) = a^x + b using given points (c, 5) and (2c, 161). This detailed step-by-step solution covers exponential functions, quadratic equations, and the quadratic formula. Suitable for Grades 10-12.

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