Sure! Let's solve the equation step-by-step.
Given the equation:
[tex]\[ 5u(u + 2)(u - 6) = 0 \][/tex]
First, we can simplify the problem by dividing both sides of the equation by 5 (since it is a non-zero constant), which doesn't affect the equality. This gives us:
[tex]\[ u(u + 2)(u - 6) = 0 \][/tex]
According to the zero-product property, if a product of several factors equals zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\( u \)[/tex]:
1. Set the first factor equal to zero:
[tex]\[ u = 0 \][/tex]
2. Set the second factor equal to zero:
[tex]\[ u + 2 = 0 \][/tex]
Solving for [tex]\( u \)[/tex], we get:
[tex]\[ u = -2 \][/tex]
3. Set the third factor equal to zero:
[tex]\[ u - 6 = 0 \][/tex]
Solving for [tex]\( u \)[/tex], we get:
[tex]\[ u = 6 \][/tex]
So, the solutions to the given equation [tex]\( 5u(u + 2)(u - 6) = 0 \)[/tex] are:
[tex]\[ u = 0, \quad u = -2, \quad \text{and} \quad u = 6 \][/tex]
