Solve the following equations
`4x^3 -13x^2 -13 x+4=0`

Answer

To solve the cubic equation \( 4x^3 - 13x^2 - 13x + 4 = 0 \), we will use the factorization method. Here’s a step-by-step solution:

### Step 1: Identify the equation
We start with the equation:
\[ 4x^3 - 13x^2 - 13x + 4 = 0 \]

### Step 2: Use the Rational Root Theorem
We will check for possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (4) divided by the factors of the leading coefficient (4). Thus, the possible rational roots are:
\[ \pm 1, \pm 2, \pm 4, \pm \frac{1}{4} \]

### Step 3: Test possible roots
Let's test \( x = -1 \):
\[ 4(-1)^3 - 13(-1)^2 - 13(-1) + 4 = 4(-1) - 13(1) + 13 + 4 = -4 - 13 + 13 + 4 = 0 \]
Since \( x = -1 \) is a root, we can factor the polynomial.

### Step 4: Factor the polynomial
Since \( x + 1 \) is a factor, we can perform polynomial long division or synthetic division to divide \( 4x^3 - 13x^2 - 13x + 4 \) by \( x + 1 \).

Using synthetic division:
- Coefficients: \( 4, -13, -13, 4 \)
- Root: \( -1 \)

Performing synthetic division:
```
-1 | 4 -13 -13 4
| -4 17 -4
---------------------
4 -17 4 0
```
The result is:
\[ 4x^2 - 17x + 4 \]
Thus, we have:
\[ 4x^3 - 13x^2 - 13x + 4 = (x + 1)(4x^2 - 17x + 4) \]

### Step 5: Solve the quadratic equation
Now we need to solve \( 4x^2 - 17x + 4 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4, b = -17, c = 4 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-17)^2 - 4 \cdot 4 \cdot 4 = 289 - 64 = 225 \]
Now substituting into the quadratic formula:
\[ x = \frac{17 \pm \sqrt{225}}{2 \cdot 4} = \frac{17 \pm 15}{8} \]
Calculating the two possible values:
1. \( x = \frac{32}{8} = 4 \)
2. \( x = \frac{2}{8} = \frac{1}{4} \)

### Step 6: Write the final solutions
The complete set of solutions for the equation \( 4x^3 - 13x^2 - 13x + 4 = 0 \) is:
\[ x = -1, \quad x = 4, \quad x = \frac{1}{4} \]

### Summary of Solutions
Thus, the solutions are:
\[ x = -1, \quad x = 4, \quad x = \frac{1}{4} \]

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