Show how many solutions are in the solution set for 2(x−5)2+2=20 .(1 point)

Question

Responses

zero
zero

infinitely many
infinitely many

two
two

one
one

Answer 1

Let's solve the equation \(2(x - 5)^2 + 2 = 20\) to determine the number of solutions in the solution set.

First, we simplify the equation:

\[2(x - 5)^2 + 2 = 20\]

Subtract 2 from both sides to isolate the quadratic term:

\[2(x - 5)^2 = 18\]

Divide by 2 to simplify further:

\[(x - 5)^2 = 9\]

Take the square root of both sides, remembering that there are two solutions to a square root (one positive and one negative):

\[x - 5 = \pm3\]

Now, solve for \(x\) by adding 5 to both sides in each case:

When the square root result is positive (+3):
\[x = 5 + 3\]
\[x = 8\]

When the square root result is negative (-3):
\[x = 5 - 3\]
\[x = 2\]

We have found two distinct solutions, \(x = 8\) and \(x = 2\), so the correct answer is:

two