Problem
If the mean of the data set \(x^2 - 7x + 10\), \(x^2 - 5x + 6\), and \(x^2 - 3x + 2\) is \(x^2 - 5x + 6\), what is the value of \(x\)?
Solution
Step 1 ：Step 1: Set up the equation for the mean of the data set. The mean is the sum of the values divided by the number of values. So we have \(\frac{x^2 - 7x + 10 + x^2 - 5x + 6 + x^2 - 3x + 2}{3} = x^2 - 5x + 6\).
Step 2 ：Step 2: Simplify the equation. Combine like terms to get \(\frac{3x^2 - 15x + 18}{3} = x^2 - 5x + 6\).
Step 3 ：Step 3: Multiply both sides by 3 to get rid of the fraction. This gives us \(3x^2 - 15x + 18 = 3x^2 - 15x + 18\).
Step 4 ：Step 4: Subtract \(3x^2 - 15x + 18\) from both sides. This gives us \(0 = 0\).
Step 5 ：Step 5: Since the equation is always true, there are infinite solutions for \(x\).
