The number of real roots of `(6 - x)^(4) + (8 - x)^(4) = 16`, is

Options

A. 0
B. 2
C. 4
D. none of these

Explanation

To solve the equation \((6 - x)^{4} + (8 - x)^{4} = 16\) and find the number of real roots, we can follow these steps:

### Step 1: Substitute \(y = 7 - x\)
We start by substituting \(y = 7 - x\). This gives us:
\[ x = 7 - y \]
Now, we can rewrite the equation in terms of \(y\):
\[ (6 - (7 - y))^{4} + (8 - (7 - y))^{4} = 16 \]
This simplifies to:
\[ (y - 1)^{4} + (y + 1)^{4} = 16 \]

### Step 2: Expand the terms
Next, we expand \((y - 1)^{4}\) and \((y + 1)^{4}\):
\[ (y - 1)^{4} = y^{4} - 4y^{3} + 6y^{2} - 4y + 1 \]
\[ (y + 1)^{4} = y^{4} + 4y^{3} + 6y^{2} + 4y + 1 \]
Adding these two expansions together:
\[ (y - 1)^{4} + (y + 1)^{4} = (y^{4} - 4y^{3} + 6y^{2} - 4y + 1) + (y^{4} + 4y^{3} + 6y^{2} + 4y + 1) \]
This simplifies to:
\[ 2y^{4} + 12y^{2} + 2 \]

### Step 3: Set the equation equal to 16
Now we set the equation equal to 16:
\[ 2y^{4} + 12y^{2} + 2 = 16 \]
Subtracting 16 from both sides gives:
\[ 2y^{4} + 12y^{2} - 14 = 0 \]
Dividing the entire equation by 2 simplifies it to:
\[ y^{4} + 6y^{2} - 7 = 0 \]

### Step 4: Let \(z = y^{2}\)
Let \(z = y^{2}\), then we rewrite the equation as:
\[ z^{2} + 6z - 7 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 6\), and \(c = -7\):
\[ z = \frac{-6 \pm \sqrt{6^{2} - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ z = \frac{-6 \pm \sqrt{36 + 28}}{2} \]
\[ z = \frac{-6 \pm \sqrt{64}}{2} \]
\[ z = \frac{-6 \pm 8}{2} \]
This gives us two solutions:
1. \(z = \frac{2}{2} = 1\)
2. \(z = \frac{-14}{2} = -7\) (not valid since \(z = y^{2} \geq 0\))

### Step 6: Find \(y\) values
Since \(z = y^{2} = 1\), we have:
\[ y = \pm 1 \]

### Step 7: Find \(x\) values
Recalling that \(y = 7 - x\):
1. If \(y = 1\), then \(7 - x = 1 \Rightarrow x = 6\)
2. If \(y = -1\), then \(7 - x = -1 \Rightarrow x = 8\)

### Conclusion
Thus, the values of \(x\) are \(6\) and \(8\), giving us a total of **2 real roots**.

### Final Answer
The number of real roots of the equation is **2**.

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