Solution: Given $ p(y) = y^2 - 6y + 9m $ and $ p(5) = 22 $, substitute $ y = 5 $: - Coaching Toolbox
Solving $ p(y) = y^2 - 6y + 9m $ Using the Given Condition $ p(5) = 22 $
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 $:
$$
p(5) = (5)^2 - 6(5) + 9m
$$
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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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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.
