Questions: Find the solution to the equation (x^3 + 10x^2 + 25x) / (x + 5) = 0

Solution Steps

First, factor the numerator \( x^{3} + 10x^{2} + 25x \). Notice that \( x \) is a common factor: \[ x^{3} + 10x^{2} + 25x = x(x^{2} + 10x + 25). \]

Next, factor the quadratic expression \( x^{2} + 10x + 25 \): \[ x^{2} + 10x + 25 = (x + 5)^{2}. \] Thus, the numerator becomes: \[ x(x + 5)^{2}. \]

Substitute the factored numerator back into the original equation: \[ \frac{x(x + 5)^{2}}{x + 5} = 0. \] Simplify the fraction by canceling out \( x + 5 \) (note that \( x \neq -5 \)): \[ x(x + 5) = 0. \]

Set each factor equal to zero: \[ x = 0 \quad \text{or} \quad x + 5 = 0. \] Solving these gives: \[ x = 0 \quad \text{or} \quad x = -5. \] However, \( x = -5 \) is excluded because it makes the denominator zero. Therefore, the only solution is: \[ x = 0. \]

Final Answer