X^4- 3x² - 4 = 0

To solve the equation \( x^4 - 3x^2 - 4 = 0 \), we can use substitution. Let \( y = x^2 \). Then, the equation becomes: \[ y^2 - 3y - 4 = 0 \] Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\). Thus, we can factor the equation as follows: \[ (y - 4)(y + 1) = 0 \] Setting each factor equal to zero gives us: 1. \( y - 4 = 0 \) \(\Rightarrow y = 4\) 2. \( y + 1 = 0 \) \(\Rightarrow y = -1\) Now, recalling that \( y = x^2 \), we substitute back: 1. For \( y = 4 \): \[ x^2 = 4 \quad \Rightarrow \quad x = \pm 2 \] 2. For \( y = -1 \): \[ x^2 = -1 \quad \Rightarrow \quad x = \pm i \] So, the complete solution for the original equation \( x^4 - 3x^2 - 4 = 0 \) is: \[ x = 2, \quad x = -2, \quad x = i, \quad x = -i \] Thus, the solutions are \( x = 2, -2, i, -i \).