To determine if the values of \(10\) and \(30\) are solutions to the equation \((x-27)(x-10)=60\), we will substitute each value into the equation and check if the equation holds.
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Substituting \(x = 10\): \[ (10-27)(10-10) = 60 \] This simplifies to: \[ (-17)(0) = 60 \] This results in: \[ 0 = 60 \] This is false. Therefore, \(10\) is not a solution.
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Substituting \(x = 30\): \[ (30-27)(30-10) = 60 \] This simplifies to: \[ (3)(20) = 60 \] This results in: \[ 60 = 60 \] This is true. Therefore, \(30\) is a solution.
In conclusion, \(10\) is not a solution, while \(30\) is a solution to the quadratic equation \((x-27)(x-10)=60\).