solve for x
1.75e^(-x*0.5) + 1.75e^(-x) + 1.75e^(-x*1.5) + 101.75e^(-x*2)=86.3995

Let's solve the equation:

\[
1.75e^{-0.5x} + 1.75e^{-x} + 1.75e^{-1.5x} + 101.75e^{-2x} = 86.3995
\]

### Step 1: Simplify the equation
First, let's factor out \( 1.75 \) from the first three terms:

\[
1.75 \left( e^{-0.5x} + e^{-x} + e^{-1.5x} \right) + 101.75e^{-2x} = 86.3995
\]

Next, divide the entire equation by 1.75 to further simplify:

\[
e^{-0.5x} + e^{-x} + e^{-1.5x} + 58.14e^{-2x} = 49.37114
\]

### Step 2: Apply substitution
Let \( y = e^{-0.5x} \), so:

\[
y^2 = e^{-x}, \quad y^3 = e^{-1.5x}, \quad y^4 = e^{-2x}
\]

Substitute into the equation:

\[
y + y^2 + y^3 + 58.14y^4 = 49.37114
\]

### Step 3: Solve the polynomial equation
This is now a polynomial equation in \( y \):

\[
58.14y^4 + y^3 + y^2 + y - 49.37114 = 0
\]

To find the value of \( y \), this equation needs to be solved using numerical methods, such as Newton's method or by using a computational tool. Once \( y \) is found, solve for \( x \) using \( y = e^{-0.5x} \).

Let's solve this numerically to find the value of \( y \).The numerical solution for \( y \) is approximately \( y \approx 0.9466 \).

### Step 4: Solve for \( x \)
Recall that \( y = e^{-0.5x} \). So,

\[
0.9466 = e^{-0.5x}
\]

Take the natural logarithm on both sides:

\[
\ln(0.9466) = -0.5x
\]

Solving for \( x \):

\[
x = -2 \ln(0.9466)
\]

Now, let's compute the value of \( x \).The solution for \( x \) is approximately \( x \approx 0.1098 \).

Would you like any further details or have any questions? Here are some related problems you might find interesting:

1. Solve a similar equation with different coefficients.
2. Explore the solution method for a non-exponential polynomial equation.
3. Analyze the behavior of the equation as one of the coefficients increases.
4. Consider the case where the constant term on the right-hand side changes.
5. Investigate the solutions for higher-order exponential equations.

**Tip:** When dealing with equations involving exponential terms, substituting variables can simplify the problem into a polynomial equation, making it easier to solve.

solve for x
1.75e^(-x*0.5) + 1.75e^(-x) + 1.75e^(-x*1.5) + 101.75e^(-x*2) = 86.3995

Learn how to solve the exponential equation 1.75e^(-x*0.5) + 1.75e^(-x) + 1.75e^(-x*1.5) + 101.75e^(-x*2) = 86.3995 using polynomial substitution and numerical methods. This detailed solution walks through algebraic simplifications and numerical techniques to solve for x. Ideal for college-level students studying advanced algebra or pre-calculus.

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