Find the roots of the equation: :$25a^2-35a+12$.
- $(5a-3)(5a-4)$
- $(5a-3)(5a+4)$
- $(5a+3)(5a-4)$
- $(5a+3)(5a+4)$
The expression factors as 25a^2 - 35a + 12 = (5a - 3)(5a - 4). Therefore, the roots are a = 3/5 and a = 4/5. The listed correct option gives the factorization rather than the roots themselves.
To find the roots of the quadratic expression 25a^2 - 35a + 12, we equate it to zero and factorize it using the middle term splitting method. We need two numbers that multiply to 300, which is the product of the leading coefficient 25 and the constant 12, and add up to the middle coefficient -35. These numbers are -15 and -20, allowing us to split the middle term and rewrite the expression as 25a^2 - 15a - 20a + 12 = 0. Factoring by grouping gives 5a(5a - 3) - 4(5a - 3) = 0, which results in the factors (5a - 3)(5a - 4) = 0.