Question

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

Answer 1

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

Only x=10 is a solution.
Only x is equal to 10 is a solution. - incorrect

Only x=30 is a solution.
Only x is equal to 30 is a solution. - no response given

Neither x=10 nor x=30 is a solution.
Neither x is equal to 10 nor x is equal to 30 is a solution. - no response given

Both x=10 and x=30 are solutions

Answer 2

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

First, let's simplify the equation:

\[
(x-27)(x-10) = 60
\]

Expanding the left-hand side:

\[
x^2 - 10x - 27x + 270 = 60
\]

Combining like terms:

\[
x^2 - 37x + 270 = 60
\]

Now, we can move the 60 to the left side:

\[
x^2 - 37x + 270 - 60 = 0
\]

This simplifies to:

\[
x^2 - 37x + 210 = 0
\]

Now, we have the quadratic equation \(x^2 - 37x + 210 = 0\). Next, we will check both values.

**Checking \(x = 10\):**

\[
10^2 - 37(10) + 210 = 0
\]
\[
100 - 370 + 210 = 0
\]
\[
100 - 370 + 210 = -70 \neq 0
\]

So \(10\) is **not** a solution.

**Checking \(x = 30\):**

\[
30^2 - 37(30) + 210 = 0
\]
\[
900 - 1110 + 210 = 0
\]
\[
900 - 1110 + 210 = 0
\]
\[
900 - 900 = 0
\]

So \(30\) **is** a solution.

In conclusion, the value 10 is **not** a solution, while the value 30 **is** a solution to the quadratic equation \((x-27)(x-10) = 60\).

Answer 3

Based on the previous calculations:

1. For \( x = 10 \):
- The equation evaluated to \(-70\), so **10 is not a solution**.

2. For \( x = 30 \):
- The equation evaluated to \(0\), so **30 is a solution**.

Therefore, the correct response is:

- **Only \( x = 30 \) is a solution.**