7. If \( \mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x}) \), solve for \( \mathrm{x}: \) \( f(x)=100^{x^{3}+x^{2}-4 x}, g(x)=10^{2 x^{2}-6 x} \)

To solve for \( x \) in the equation \( f(x) = g(x) \) where \( f(x) = 100^{x^{3}+x^{2}-4x} \) and \( g(x) = 10^{2x^{2}-6x} \), we start by rewriting \( f(x) \): Since \( 100 = 10^2 \), we can express \( f(x) \) as: \[ f(x) = (10^2)^{x^{3}+x^{2}-4x} = 10^{2(x^{3}+x^{2}-4x)} = 10^{2x^{3}+2x^{2}-8x} \] Now, we have: \[ 10^{2x^{3}+2x^{2}-8x} = 10^{2x^{2}-6x} \] Since their bases are the same, we can set the exponents equal to each other: \[ 2x^{3} + 2x^{2} - 8x = 2x^{2} - 6x \] Next, we simplify this equation. Start by subtracting \( 2x^{2} \) from both sides: \[ 2x^{3} - 8x = -6x \] Adding \( 6x \) to both sides gives: \[ 2x^{3} - 8x + 6x = 0 \] This simplifies to: \[ 2x^{3} - 2x = 0 \] Factoring out \( 2x \): \[ 2x(x^{2} - 1) = 0 \] Setting each factor to zero leads us to the solutions: 1. \( 2x = 0 \) which gives \( x = 0 \) 2. \( x^{2}-1 = 0 \) which factors to \( (x-1)(x+1) = 0 \) yielding \( x = 1 \) and \( x = -1 \) Thus, the complete solution set is: \[ x = 0, 1, -1 \]