Using the properties of proportion, solve for x, given
`(x^(4)+1)/(2x^(2))=17/(8)`.

Answer

To solve the equation \(\frac{x^4 + 1}{2x^2} = \frac{17}{8}\) using the properties of proportion, we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:
\[ 8(x^4 + 1) = 34x^2 \]

### Step 2: Expand and Rearrange
Expanding the left side gives:
\[ 8x^4 + 8 = 34x^2 \]
Now, rearranging the equation to set it to zero:
\[ 8x^4 - 34x^2 + 8 = 0 \]

### Step 3: Substitute \(y = x^2\)
Let \(y = x^2\). Then, the equation becomes:
\[ 8y^2 - 34y + 8 = 0 \]

### Step 4: Use the Quadratic Formula
We will use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 8\), \(b = -34\), and \(c = 8\):
\[ y = \frac{34 \pm \sqrt{(-34)^2 - 4 \cdot 8 \cdot 8}}{2 \cdot 8} \]
Calculating the discriminant:
\[ (-34)^2 = 1156 \]
\[ 4 \cdot 8 \cdot 8 = 256 \]
Thus,
\[ b^2 - 4ac = 1156 - 256 = 900 \]
Now substituting back into the formula:
\[ y = \frac{34 \pm \sqrt{900}}{16} \]
\[ y = \frac{34 \pm 30}{16} \]

### Step 5: Solve for \(y\)
Calculating the two possible values for \(y\):
1. \(y = \frac{64}{16} = 4\)
2. \(y = \frac{4}{16} = \frac{1}{4}\)

### Step 6: Back Substitute for \(x\)
Since \(y = x^2\), we have:
1. \(x^2 = 4 \Rightarrow x = \pm 2\)
2. \(x^2 = \frac{1}{4} \Rightarrow x = \pm \frac{1}{2}\)

### Final Solution
Thus, the complete set of solutions for \(x\) is:
\[ x = 2, -2, \frac{1}{2}, -\frac{1}{2} \]

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