Solution: Given $ p(y) = y^2 - 6y + 9m $ and $ p(5) = 22 $, substitute $ y = 5 $:
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 $.
Step 1: Substitute $ y = 5 $ into the function
Given: $$ p(y) = y^2 - 6y + 9m $$
Substitute $ y = 5 $: $$ p(5) = (5)^2 - 6(5) + 9m $$
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: $$ -5 + 9m = 22 $$
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.
