Question

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

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

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

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

Only x=10
is a solution.

Answer 1

To determine if the values 10 and 30 are solutions to the quadratic equation \((x−27)(x−10)=60\), we can start by rearranging the equation to standard form.

1. Expand the left side:
\[
(x - 27)(x - 10) = x^2 - 10x - 27x + 270 = x^2 - 37x + 270
\]
So the equation becomes:
\[
x^2 - 37x + 270 = 60
\]

2. Subtract 60 from both sides:
\[
x^2 - 37x + 210 = 0
\]

Now, we can check if \(x = 10\) and \(x = 30\) are solutions by substituting these values into the quadratic equation \(x^2 - 37x + 210 = 0\):

**Checking \(x = 10\):**
\[
10^2 - 37(10) + 210 = 100 - 370 + 210 = -60 \text{ (not equal to 0)}
\]

**Checking \(x = 30\):**
\[
30^2 - 37(30) + 210 = 900 - 1110 + 210 = 0 \text{ (equal to 0)}
\]

From the calculations:
- \(x = 10\) is **not** a solution.
- \(x = 30\) **is** a solution.

Therefore, only \(x = 30\) is a solution.

The correct response is:
"Only \(x = 30\) is a solution."