The solutions of the quadratic equation `x^(4)-26x^(2)+25=0` are

Options

A. 5,3
B. `-1,4`
C. `+-1,+-5`
D. `-5,1`

Answer

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

### Step 1: Substitute \( t = x^2 \)
We can rewrite the equation in terms of \( t \):
\[ t^2 - 26t + 25 = 0 \]

### Step 2: Factor the quadratic equation
Next, we will factor the quadratic equation \( t^2 - 26t + 25 \). We need two numbers that multiply to \( 25 \) and add up to \( -26 \). The numbers are \( -25 \) and \( -1 \):
\[ (t - 25)(t - 1) = 0 \]

### Step 3: Solve for \( t \)
Setting each factor equal to zero gives us:
\[ t - 25 = 0 \quad \Rightarrow \quad t = 25 \]
\[ t - 1 = 0 \quad \Rightarrow \quad t = 1 \]

### Step 4: Substitute back to find \( x \)
Since we substituted \( t = x^2 \), we now substitute back to find \( x \):
1. For \( t = 25 \):
\[ x^2 = 25 \quad \Rightarrow \quad x = \pm 5 \]
2. For \( t = 1 \):
\[ x^2 = 1 \quad \Rightarrow \quad x = \pm 1 \]

### Step 5: List all solutions
Combining the solutions from both cases, we have:
\[ x = 5, -5, 1, -1 \]

### Final Answer:
The solutions of the quadratic equation \( x^4 - 26x^2 + 25 = 0 \) are:
\[ x = 5, -5, 1, -1 \]

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