To solve the equation [tex]\(25m^2 = 121\)[/tex], let's follow these detailed steps:
1. Rearrange the Equation:
[tex]\[ 25m^2 - 121 = 0 \][/tex]
2. Factor the Equation:
We notice that the left side of the equation is a difference of squares. A difference of squares can be factored as follows:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In our case:
[tex]\[ 25m^2 - 121 = (5m)^2 - 11^2 \][/tex]
This can be factored into:
[tex]\[ (5m - 11)(5m + 11) = 0 \][/tex]
3. Set Each Factor to Zero:
For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor to zero and solve for [tex]\( m \)[/tex]:
[tex]\[ 5m - 11 = 0 \][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[ 5m = 11 \][/tex]
[tex]\[ m = \frac{11}{5} \][/tex]
and
[tex]\[ 5m + 11 = 0 \][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[ 5m = -11 \][/tex]
[tex]\[ m = \frac{-11}{5} \][/tex]
4. Write the Solutions:
So, the solutions to the equation [tex]\( 25m^2 = 121 \)[/tex] are:
[tex]\[ m = \frac{11}{5}, \frac{-11}{5} \][/tex]
Hence, the solutions are:
[tex]\[ \boxed{\frac{-11}{5}, \frac{11}{5}} \][/tex]
