Question

Determine whether the values 10 and 30 are solutions to the quadratic equation (x−27)(x−10)=60
.(1 point)
Responses

Only x=30
is a solution.
Only x equals 30 is a solution.

Only x=10
is a solution.
Only x equals 10 is a solution.

Both x=10
and x=30
are solutions.
Both x equals 10 and x equals 30 are solutions.

Neither x=10
nor x=30
is a solution.

Answer 1

To determine whether the values 10 and 30 are solutions to the quadratic equation \((x−27)(x−10)=60\), we can substitute each value into the equation and check if it holds true.

1. **Checking \(x = 10\)**:
\[
(10 - 27)(10 - 10) = (10 - 27)(0) = (-17)(0) = 0
\]
We need to check if \(0 = 60\), which is false. So, \(x = 10\) is **not a solution**.

2. **Checking \(x = 30\)**:
\[
(30 - 27)(30 - 10) = (3)(20) = 60
\]
We need to check if \(60 = 60\), which is true. So, \(x = 30\) **is a solution**.

From the checks above, the only value that satisfies the equation is \(x = 30\), while \(x = 10\) does not.

Thus, the correct response is: **Only x=30 is a solution.**