Math Problem Statement

Evaluate the equation 2f = (9 - x^3) / 4 for the values x = -2, 0, 3, and 6.

Solution

To solve the equation 2f=9x342f = \frac{9 - x^3}{4} for the given values of x=2,0,3,x = -2, 0, 3, and 66, we will follow these steps:

  1. Substitute each value of xx into the equation to solve for ff.
  2. Simplify the equation to find the value of ff.

Step 1: Solve for ff when x=2x = -2

Substitute x=2x = -2 into the equation: 2f=9(2)342f = \frac{9 - (-2)^3}{4} 2f=9(8)4=9+84=1742f = \frac{9 - (-8)}{4} = \frac{9 + 8}{4} = \frac{17}{4} f=178f = \frac{17}{8}

Step 2: Solve for ff when x=0x = 0

Substitute x=0x = 0 into the equation: 2f=90342f = \frac{9 - 0^3}{4} 2f=904=942f = \frac{9 - 0}{4} = \frac{9}{4} f=98f = \frac{9}{8}

Step 3: Solve for ff when x=3x = 3

Substitute x=3x = 3 into the equation: 2f=93342f = \frac{9 - 3^3}{4} 2f=9274=184=922f = \frac{9 - 27}{4} = \frac{-18}{4} = -\frac{9}{2} f=94f = -\frac{9}{4}

Step 4: Solve for ff when x=6x = 6

Substitute x=6x = 6 into the equation: 2f=96342f = \frac{9 - 6^3}{4} 2f=92164=20742f = \frac{9 - 216}{4} = \frac{-207}{4} f=2078f = -\frac{207}{8}

Final Results:

  • x=2x = -2, f=178f = \frac{17}{8}
  • x=0x = 0, f=98f = \frac{9}{8}
  • x=3x = 3, f=94f = -\frac{9}{4}
  • x=6x = 6, f=2078f = -\frac{207}{8}

Would you like more details or have any questions?

Here are 5 relative questions:

  1. What happens if xx is a non-integer, such as x=2.5x = 2.5?
  2. How would the equation change if the coefficient of ff was 3 instead of 2?
  3. Can ff ever be zero? Under what conditions?
  4. What is the behavior of f(x)f(x) as xx becomes very large or very small?
  5. How does changing the constant 9 in the equation affect the values of ff?

Tip: When substituting values into equations, make sure to correctly handle negative signs and powers, as they can significantly change the outcome.

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Math Problem Analysis

Mathematical Concepts

Algebra
Equations
Substitution

Formulas

2f = (9 - x^3) / 4

Theorems

Basic Algebraic Operations

Suitable Grade Level

Grades 8-10