Solving $ p(y) = y^2 - 6y + 9m $ Using the Given Condition $ p(5) = 22 $

When solving for unknown parameters in quadratic functions, substitution is one of the most effective techniques. In this article, we explain the step-by-step solution to determine the value of $ m $ in the function $ p(y) = y^2 - 6y + 9m $, using the condition that $ p(5) = 22 $.

Understanding the Context

Step 1: Substitute $ y = 5 $ into the function

Given:
$$
p(y) = y^2 - 6y + 9m
$$

Substitute $ y = 5 $:
$$
p(5) = (5)^2 - 6(5) + 9m
$$

Solution: Given $ p(y) = y^2 - 6y + 9m $ and $ p(5) = 22 $, substitute $ y = 5 $:

Key Insights

Step 2: Simplify the expression

Compute each term:
$$
p(5) = 25 - 30 + 9m
$$
$$
p(5) = -5 + 9m
$$

Step 3: Apply the given condition

We’re told that $ p(5) = 22 $. So:
$$
-5 + 9m = 22
$$

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Final Thoughts

Step 4: Solve for $ m $

Add 5 to both sides:
$$
9m = 27
$$

Divide by 9:
$$
m = 3
$$

Final Result

The value of $ m $ that satisfies $ p(y) = y^2 - 6y + 9m $ and the condition $ p(5) = 22 $ is:
$$
oxed{3}
$$

This method is essential in algebra for diagnosing quadratic functions and solving for parameters directly from function values — a key skill in school math and standardized problem-solving.