Question
What value of c makes the polynomial below a perfect square?
x^2 + 14x+c
C =
Answer
÷49
# 28
:: 196
÷7
x^2 + 14x+c
C =
Answer
÷49
# 28
:: 196
÷7
Answers
Answered by
GPT-4o mini
To make the polynomial \( x^2 + 14x + c \) a perfect square, we want it to be expressible in the form \( (x + a)^2 \) for some value \( a \).
The general expansion of \( (x + a)^2 \) is:
\[
(x + a)^2 = x^2 + 2ax + a^2
\]
In our polynomial \( x^2 + 14x + c \), we can compare the coefficients:
1. The coefficient of \( x \) gives us \( 2a = 14 \).
2. Solving for \( a \) gives us:
\[
a = \frac{14}{2} = 7
\]
3. To find \( c \), we use \( a^2 \):
\[
c = a^2 = 7^2 = 49
\]
Thus, the value of \( c \) that makes the polynomial \( x^2 + 14x + c \) a perfect square is:
\[
\boxed{49}
\]
The general expansion of \( (x + a)^2 \) is:
\[
(x + a)^2 = x^2 + 2ax + a^2
\]
In our polynomial \( x^2 + 14x + c \), we can compare the coefficients:
1. The coefficient of \( x \) gives us \( 2a = 14 \).
2. Solving for \( a \) gives us:
\[
a = \frac{14}{2} = 7
\]
3. To find \( c \), we use \( a^2 \):
\[
c = a^2 = 7^2 = 49
\]
Thus, the value of \( c \) that makes the polynomial \( x^2 + 14x + c \) a perfect square is:
\[
\boxed{49}
\]