To identify the equations that can be used to find the zeros of the quadratic expression [tex]\(-16t^2 + 64t\)[/tex], we need to set this expression equal to zero and solve for [tex]\(t\)[/tex]. Let's go through the steps:
1. Set the quadratic expression equal to zero:
[tex]\[ -16t^2 + 64t = 0 \][/tex]
2. Factor the expression:
Notice that we can factor out a common term [tex]\( -16t \)[/tex] from both parts of the expression:
[tex]\[ -16t(t - 4) = 0 \][/tex]
3. Set each factor equal to zero:
For the equation [tex]\( -16t(t - 4) = 0 \)[/tex], according to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So we set each factor to zero:
[tex]\[ -16t = 0 \][/tex]
[tex]\[ t - 4 = 0 \][/tex]
4. Solve each equation:
- For [tex]\( -16t = 0 \)[/tex]:
[tex]\[ t = 0 \][/tex]
- For [tex]\( t - 4 = 0 \)[/tex]:
[tex]\[ t = 4 \][/tex]
So, the equations used to find the zeros of the expression [tex]\(-16t^2 + 64t\)[/tex] are:
[tex]\[ -16t = 0 \quad \text{and} \quad t - 4 = 0 \][/tex]
Therefore, the correct option among the given choices is:
[tex]\[ -16t = 0 \quad \text{and} \quad t - 4 = 0 \][/tex]
