Find the value of \(c\) such that each expression is a perfect-square trinomial. Please show all steps.
1. \(x^2 - 16x + c\)
2. \(p^2 - 14p + c\)
3. \(b^2 + 18b + c\)
4. \(n^2 - n + c\)
Answer (2)
( a ± b ) 2 = a 2 ± 2 ab + b 2 1. x 2 − 16 x + c = x 2 − 2 x ⋅ 8 + c → c = 8 2 = 64 x 2 − 16 x + 64 = x 2 − 2 x ⋅ 8 + 8 2 = ( x − 8 ) 2 2. p 2 − 14 p + c = p 2 − 2 p ⋅ 7 + c → c = 7 2 = 49 p 2 − 14 p + 49 = p 2 − 2 p ⋅ 7 + 7 2 = ( p − 7 ) 2
3. b 2 + 18 b + c = b 2 + 2 b ⋅ 9 + c → c = 9 2 = 81 b 2 + 18 b + 81 = b 2 + 2 b ⋅ 9 + 9 2 = ( b + 9 ) 2 4. n 2 − n + c = n 2 − 2 n ⋅ 2 1 → c = ( 2 1 ) 2 = 4 1 2 − n + 4 1 = n 2 − 2 n ⋅ 2 1 + ( 2 1 ) 2 = ( n − 2 1 ) 2
To find c for each expression to be a perfect-square trinomial, we derive c as follows: for x 2 − 16 x + c , c = 64 ; for p 2 − 14 p + c , c = 49 ; for b 2 + 18 b + c , c = 81 ; and for n 2 − n + c , c = 4 1 .
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