To determine how many solutions are in the solution set for the equation \(2(x - 5)^2 + 2 = 20\), we can start by simplifying it step-by-step.
-
Subtract 2 from both sides of the equation: \[ 2(x - 5)^2 + 2 - 2 = 20 - 2 \] This simplifies to: \[ 2(x - 5)^2 = 18 \]
-
Divide both sides by 2: \[ (x - 5)^2 = 9 \]
-
Take the square root of both sides: \[ x - 5 = \pm 3 \]
This leads to two equations:
- \(x - 5 = 3\)
- \(x - 5 = -3\)
- Solve each equation:
From \(x - 5 = 3\): \[ x = 3 + 5 = 8 \]
From \(x - 5 = -3\): \[ x = -3 + 5 = 2 \]
Thus, the solutions are \(x = 8\) and \(x = 2\).
Since we found two distinct solutions, the number of solutions in the solution set is two.
1 answer