What is the solution set for the equation `x^(4) - 26 x^(2) + 25 = 0` ?

Options

A. {-5,-1,1,5}
B. {-5,-1}
C. {1,5}
D. {-5,0,1,5}

Answer

To solve the equation \( x^4 - 26x^2 + 25 = 0 \), we can start by making a substitution to simplify the equation. Let's set \( y = x^2 \). Then, the equation becomes:

\[ y^2 - 26y + 25 = 0 \]

### Step 1: Factor the quadratic equation
Next, we will factor the quadratic equation \( y^2 - 26y + 25 \). We need to find two numbers that multiply to \( 25 \) (the constant term) and add up to \( -26 \) (the coefficient of \( y \)). The numbers that satisfy this are \( -25 \) and \( -1 \).

Thus, we can write:

\[ (y - 25)(y - 1) = 0 \]

### Step 2: Solve for \( y \)
Now, we set each factor equal to zero:

1. \( y - 25 = 0 \) → \( y = 25 \)
2. \( y - 1 = 0 \) → \( y = 1 \)

### Step 3: Substitute back for \( x \)
Since we made the substitution \( y = x^2 \), we now substitute back to find \( x \):

1. For \( y = 25 \):
\[ x^2 = 25 \implies x = \pm 5 \]

2. For \( y = 1 \):
\[ x^2 = 1 \implies x = \pm 1 \]

### Step 4: Write the solution set
Combining all the solutions, we have:

\[ x = -5, -1, 1, 5 \]

Thus, the solution set is:

\[ \{-5, -1, 1, 5\} \]

### Final Answer
The solution set for the equation \( x^4 - 26x^2 + 25 = 0 \) is:

\[ \{-5, -1, 1, 5\} \]