To solve the equation [tex]\((y-5)(y+5)=0\)[/tex] for [tex]\(y\)[/tex], we need to use the property of zero products, which states that if the product of two factors equals zero, then at least one of the factors must be zero. Here are the steps:
1. Identify the factors:
The given equation is already in factored form: [tex]\((y-5)(y+5)=0\)[/tex].
2. Apply the zero product property:
According to this property, we set each factor equal to zero and solve for [tex]\(y\)[/tex]:
- First factor: [tex]\(y - 5 = 0\)[/tex]
- Second factor: [tex]\(y + 5 = 0\)[/tex]
3. Solve each equation separately:
- For the first factor [tex]\(y - 5 = 0\)[/tex]:
[tex]\[
y - 5 = 0 \implies y = 5
\][/tex]
- For the second factor [tex]\(y + 5 = 0\)[/tex]:
[tex]\[
y + 5 = 0 \implies y = -5
\][/tex]
4. List the solutions:
The solutions to the equation [tex]\((y-5)(y+5)=0\)[/tex] are therefore:
[tex]\[
y = 5 \quad \text{and} \quad y = -5
\][/tex]
So, the values of [tex]\(y\)[/tex] that satisfy the given equation are [tex]\(y = 5\)[/tex] and [tex]\(y = -5\)[/tex].
