If `4x + (1)/(x) = 5, x ne 0,` then the value of `(5x )/(4x ^(2) + 10 x + 1)` is

Options

A. `1/2`
B. `1/3`
C. `2/3`
D. `3`

Answer

To solve the equation \( 4x + \frac{1}{x} = 5 \) and find the value of \( \frac{5x}{4x^2 + 10x + 1} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ 4x + \frac{1}{x} = 5 \]
To eliminate the fraction, we can multiply both sides by \( x \) (since \( x \neq 0 \)):
\[ 4x^2 + 1 = 5x \]

### Step 2: Rearrange the equation
Now, we can rearrange this equation to form a standard quadratic equation:
\[ 4x^2 - 5x + 1 = 0 \]

### Step 3: Solve the quadratic equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4 \), \( b = -5 \), and \( c = 1 \):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 4 \cdot 1 = 25 - 16 = 9 \]
Now, substituting back into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{9}}{2 \cdot 4} = \frac{5 \pm 3}{8} \]
This gives us two potential solutions:
\[ x = \frac{8}{8} = 1 \quad \text{and} \quad x = \frac{2}{8} = \frac{1}{4} \]

### Step 4: Substitute \( x \) into the expression
Next, we need to find the value of \( \frac{5x}{4x^2 + 10x + 1} \). We will evaluate this for both values of \( x \).

#### For \( x = 1 \):
\[ 4(1)^2 + 10(1) + 1 = 4 + 10 + 1 = 15 \]
Thus,
\[ \frac{5(1)}{15} = \frac{5}{15} = \frac{1}{3} \]

#### For \( x = \frac{1}{4} \):
\[ 4\left(\frac{1}{4}\right)^2 + 10\left(\frac{1}{4}\right) + 1 = 4 \cdot \frac{1}{16} + \frac{10}{4} + 1 = \frac{1}{4} + \frac{5}{2} + 1 \]
Converting \( \frac{5}{2} \) and \( 1 \) to quarters:
\[ \frac{1}{4} + \frac{10}{4} + \frac{4}{4} = \frac{1 + 10 + 4}{4} = \frac{15}{4} \]
Thus,
\[ \frac{5\left(\frac{1}{4}\right)}{\frac{15}{4}} = \frac{\frac{5}{4}}{\frac{15}{4}} = \frac{5}{15} = \frac{1}{3} \]

### Conclusion
In both cases, we find that:
\[ \frac{5x}{4x^2 + 10x + 1} = \frac{1}{3} \]

### Final Answer
The value of \( \frac{5x}{4x^2 + 10x + 1} \) is \( \frac{1}{3} \).

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