X²(x+5)=24

To solve the equation \( x^2(x + 5) = 24 \), we start by rearranging it into standard form: 1. First, expand the left side: \[ x^3 + 5x^2 = 24 \] 2. Subtract 24 from both sides: \[ x^3 + 5x^2 - 24 = 0 \] Now we need to solve the cubic equation \( x^3 + 5x^2 - 24 = 0 \). To find the possible rational roots, we can use the Rational Root Theorem, which suggests that the possible rational roots are factors of the constant term (-24) divided by factors of the leading coefficient (1). So, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \). Let's test some of these possible roots. Starting with \( x = 2 \): \[ 2^3 + 5(2^2) - 24 = 8 + 20 - 24 = 4 \quad \text{(not a root)} \] Next, try \( x = -3 \): \[ (-3)^3 + 5(-3)^2 - 24 = -27 + 45 - 24 = -6 \quad \text{(not a root)} \] Next, try \( x = 1 \): \[ 1^3 + 5(1^2) - 24 = 1 + 5 - 24 = -18 \quad \text{(not a root)} \] Next, try \( x = -4 \): \[ (-4)^3 + 5(-4)^2 - 24 = -64 + 80 - 24 = -8 \quad \text{(not a root)} \] Next, try \( x = 3 \): \[ 3^3 + 5(3^2) - 24 = 27 + 45 - 24 = 48 \quad \text{(not a root)} \] Next, try \( x = -2 \): \[ (-2)^3 + 5(-2)^2 - 24 = -8 + 20 - 24 = -12 \quad \text{(not a root)} \] Next, try \( x = -1 \): \[ (-1)^3 + 5(-1)^2 - 24 = -1 + 5 - 24 = -20 \quad \text{(not a root)} \] Next, try \( x = 4 \): \[ 4^3 + 5(4^2) - 24 = 64 + 80 - 24 = 120 \quad \text{(not a root)} \] Next, try \( x = -5 \): \[ (-5)^3 + 5(-5)^2 - 24 = -125 + 125 - 24 = -24 \quad \text{(not a root)} \] Next, try \( x = 2 \): \[ 2^3 + 5(2^2) - 24 = 8 + 20 - 24 = 4 \quad \text{(not a root)} \] Finally, let’s try \( x = 3 \): \[ 3^3 + 5(3^2) - 24 = 27 + 45 - 24 = 48 \quad \text{(not a root)} \] Now, since the rational roots did not yield a solution, we can try using numerical methods or graphical solutions to estimate roots. Using techniques like Descartes' Rule of Signs or other polynomial root-finding algorithms (or graphing the function), we may find approximate or exact roots. You can also apply numerical methods such as Newton's method or the bisection method, or use software/calculators to find roots. Would you like to proceed with a numerical solution method or explore another approach?