_5X⁴+30X²-25=0

To solve the equation \( -5X^4 + 30X^2 - 25 = 0 \), we can simplify it by dividing all terms by \(-5\): \[ X^4 - 6X^2 + 5 = 0 \] Next, we can use a substitution to make this a quadratic equation. Let \( Y = X^2 \). Then, the equation transforms into: \[ Y^2 - 6Y + 5 = 0 \] Now, we can factor the quadratic: \[ (Y - 1)(Y - 5) = 0 \] Setting each factor equal to zero gives: 1. \( Y - 1 = 0 \) → \( Y = 1 \) 2. \( Y - 5 = 0 \) → \( Y = 5 \) Now, recall that \( Y = X^2 \). We substitute back to find \( X \): 1. For \( Y = 1 \): \[ X^2 = 1 \implies X = 1 \quad \text{or} \quad X = -1 \] 2. For \( Y = 5 \): \[ X^2 = 5 \implies X = \sqrt{5} \quad \text{or} \quad X = -\sqrt{5} \] Thus, the complete set of solutions for the original equation is: \[ X = 1, \quad X = -1, \quad X = \sqrt{5}, \quad X = -\sqrt{5} \]