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We want to solve the following equation.

\[x^3-4x=2^x\]
Three of the solutions are
\[x\approx -2.0\],
\[x\approx-0.2\], and
\[x\approx 9.8\].
Find the other solution.
Hint: Use a graphing calculator.
Round your answer to the nearest tenth.

1 answer

To solve the equation

\[ x^3 - 4x = 2^x \]

we can rewrite it as

\[ f(x) = x^3 - 4x - 2^x = 0. \]

We know three solutions:

  • \( x \approx -2.0 \)
  • \( x \approx -0.2 \)
  • \( x \approx 9.8 \)

To find the other solution, we can check the behavior of the function \( f(x) \) and look for additional intersections between \( f(x) \) and the x-axis.

Using a graphing calculator or computational tool, we can plot the function \( f(x) \).

  1. Start by evaluating the behavior of \( f(x) \):

    • As \( x \) becomes very negative, \( 2^x \) approaches 0, and \( f(x) \) behaves like \( x^3 \) (which is negative).
    • As \( x \) increases, \( 2^x \) grows exponentially compared to the polynomial \( x^3 - 4x \).

  2. Look for intersections:

    • First, evaluate points around \( x = 0 \):
      • \( f(-1) = (-1)^3 - 4(-1) - 2^{-1} = -1 + 4 - 0.5 = 2.5 \) (positive)
      • \( f(0) = 0 - 0 - 1 = -1 \) (negative)
      • \( f(1) = 1 - 4 - 2 = -5 \) (negative)
      • \( f(2) = 8 - 8 - 4 = -4 \) (negative)
      • \( f(3) = 27 - 12 - 8 = 7 \) (positive)

This tells us there is likely a root between \( x = 0 \) and \( x = 3 \).

  1. Narrow down further:
    • Check \( f(2.5) \): \[ f(2.5) = (2.5)^3 - 4(2.5) - 2^{2.5} \approx 15.625 - 10 - 5.6569 \approx 0.968 \text{ (positive)} \]
    • Check \( f(2.2) \): \[ f(2.2) = (2.2)^3 - 4(2.2) - 2^{2.2} \approx 10.648 - 8.8 - 4.594 = -2.746 \text{ (negative)} \]
    • Check \( f(2.4) \): \[ f(2.4) = (2.4)^3 - 4(2.4) - 2^{2.4} \approx 13.824 - 9.6 - 4.727 = -0.503 \text{ (negative)} \]
    • Check \( f(2.3) \): \[ f(2.3) = (2.3)^3 - 4(2.3) - 2^{2.3} \approx 12.167 - 9.2 - 4.924 = -2.957 \text{ (negative)} \]
    • Check \( f(2.1) \): \[ f(2.1) = (2.1)^3 - 4(2.1) - 2^{2.1} \approx 9.261 - 8.4 - 4.287 = -3.426 \text{ (negative)} \]
    • Check \( f(2.6) \): \[ f(2.6) = (2.6)^3 - 4(2.6) - 2^{2.6} \approx 17.576 - 10.4 - 6.348 = 0.828 \text{ (positive)} \]

Now, we have determined that there is indeed a zero between \( x=2.4 \) and \( x=2.6 \).

Using further numerical or graphical methods, we check:

  • \( x \approx 2.5 \) gives us 0.968.
  • With a binary search or graphing tool, we find the zero is approximately around \( 2.5 \).

Final result:

Rounding to the nearest tenth, the other solution to the equation \( x^3 - 4x = 2^x \) is:

\[ \boxed{2.5}. \]