To form the perfect square trinomial from the given equation \(x^2 + 3x + c = \frac{7}{4} + c\), we need to complete the square. Here's a detailed, step-by-step process to find the value of \(c\):
1. Identify the Coefficient of the Linear Term:
The linear term in the expression is \(3x\). The coefficient \(B\) in this term is 3.
2. Halve the Coefficient of the Linear Term:
To complete the square, we take half of the coefficient of \(x\). Thus, we have:
[tex]\[
\frac{B}{2} = \frac{3}{2}
\][/tex]
3. Square the Result of Halving the Coefficient:
Next, we square the result obtained above:
[tex]\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\][/tex]
4. Value of \(c\):
The constant term \(c\) that completes the square is the squared result from the previous step:
[tex]\[
c = \frac{9}{4}
\][/tex]
Therefore, the value of \(c\) that forms a perfect square trinomial is:
\(
\boxed{2.25}
\) or \(2.25 = \frac{9}{4}\).
In conclusion, by following the method of completing the square, the required value of [tex]\(c\)[/tex] is [tex]\(\boxed{2.25}\)[/tex].
