Questions: Find all values of c for which (g(c)) is the indicated value. [g(x)=fracx2+frac4x ; g(c)=3] c=. (Simplify your answer. Use a comma to separate answers as needed.)

$Find all values of c for which (g(c)) is the indicated value. [g(x)=fracx2+frac4x ; g(c)=3] c=. (Simplify your answer. Use a comma to separate answers as needed.)$

Transcript text: Find all values of c for which $\mathrm{g}(\mathrm{c})$ is the indicated value. \[ \begin{array}{l} g(x)=\frac{x}{2}+\frac{4}{x} ; g(c)=3 \\ c=\square . \end{array} \] (Simplify your answer. Use a comma to separate answers as needed.)

Solution

Solution Steps

To find the values of $ c $ for which $ g(c) = 3 $, we need to solve the equation $ \frac{c}{2} + \frac{4}{c} = 3 $. This involves multiplying through by $ c $ to clear the fraction, forming a quadratic equation, and then solving for $ c $.

Step 1: Set Up the Equation

We start with the function $ g(x) = \frac{x}{2} + \frac{4}{x} $ and set it equal to 3 to find $ c $: \[ g(c) = 3 \implies \frac{c}{2} + \frac{4}{c} = 3 \]

Step 2: Clear the Fraction

To eliminate the fraction, we multiply both sides by $ c $: \[ c \left( \frac{c}{2} + \frac{4}{c} \right) = 3c \implies \frac{c^2}{2} + 4 = 3c \]

Step 3: Rearrange to Form a Quadratic Equation

Rearranging the equation gives us: \[ \frac{c^2}{2} - 3c + 4 = 0 \] Multiplying through by 2 to eliminate the fraction results in: \[ c^2 - 6c + 8 = 0 \]

Step 4: Solve the Quadratic Equation

Factoring the quadratic equation, we find: \[ (c - 2)(c - 4) = 0 \] Thus, the solutions are: \[ c = 2 \quad \text{and} \quad c = 4 \]