Find an equivalent equation to x^2 − 6x + 5 = 0 (same solution set): Choose the equation that has exactly the same set of real solutions as x^2 − 6x + 5 = 0.

Introduction / Context:Two equations are equivalent (have the same solution set) if each solution to one satisfies the other and vice versa. The quadratic x^2 − 6x + 5 = 0 factors neatly, so we can convert it to an absolute-value statement describing the same two points on the number line.

Given Data / Assumptions:

• Original: x^2 − 6x + 5 = 0
• We need an equation with exactly the same real roots.

Concept / Approach:Factor first: x^2 − 6x + 5 = (x − 1)(x − 5) = 0 ⇒ x = 1 or x = 5. The midpoint of the roots is 3 and each root lies 2 units from 3. That fact is captured by |x − 3| = 2.

Step-by-Step Solution:

Verification / Alternative check:Plug x = 1 and x = 5 into |x − 3| = 2: both satisfy it. The other polynomial options have different root sets (e.g., x^2 − 5x + 6 has roots 2 and 3).

Why Other Options Are Wrong:They yield different roots: coefficient changes alter the solution set. Only the absolute-value equation encodes exactly the same two points.

Common Pitfalls:Assuming multiplying by a constant keeps roots when constants shift the equation incorrectly; equivalence requires identical solution sets.

Final Answer:|x − 3| = 2