Let's consider the quadratic expression [tex]\( x^2 - 7x + c \)[/tex]. To determine the value of [tex]\( c \)[/tex] that will make this expression a perfect square trinomial, we need to use the property of perfect square trinomials. A perfect square trinomial in the form [tex]\( x^2 + bx + c \)[/tex] has [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex].
1. In the given expression [tex]\( x^2 - 7x + c \)[/tex], identify the coefficient [tex]\( b \)[/tex] of the linear term [tex]\( x \)[/tex]. Here, [tex]\( b = -7 \)[/tex].
2. To form a perfect square trinomial, calculate [tex]\( c \)[/tex] using the formula [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex].
3. Plug [tex]\( b = -7 \)[/tex] into the formula:
[tex]\[
c = \left(\frac{-7}{2}\right)^2
\][/tex]
4. Evaluate the expression inside the parentheses first:
[tex]\[
\frac{-7}{2} = -3.5
\][/tex]
5. Square the result:
[tex]\[
c = (-3.5)^2 = 12.25
\][/tex]
Thus, the value of [tex]\( c \)[/tex] that makes the expression [tex]\( x^2 - 7x + c \)[/tex] a perfect square trinomial is [tex]\( \boxed{12.25} \)[/tex].
To match this result with the given choices, note that:
[tex]\[
\frac{49}{4} = 12.25
\][/tex]
Therefore, the correct choice is [tex]\( \frac{49}{4} \)[/tex].
So, the value of [tex]\( c \)[/tex] that makes [tex]\( x^2 - 7x + c \)[/tex] a perfect square trinomial is:
[tex]\(
\boxed{\frac{49}{4}}
\)[/tex]
