Certainly! Let's solve the given problem step-by-step and form a perfect square trinomial while finding the value of [tex]\( c \)[/tex].
We start with the equation:
[tex]\[ x^2 + 3x + c = \frac{7}{4} + c \][/tex]
Our goal is to form a perfect square trinomial on the left-hand side. A perfect square trinomial can be written in the form [tex]\((x+b)^2\)[/tex], where [tex]\( b \)[/tex] is determined by the coefficient of [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Identify the Coefficient of [tex]\( x \)[/tex]:
The coefficient of [tex]\( x \)[/tex] in the equation [tex]\( x^2 + 3x + c \)[/tex] is [tex]\( 3 \)[/tex].
2. Determine [tex]\( b \)[/tex]:
To form a perfect square trinomial, we take half of the coefficient of [tex]\( x \)[/tex]. So,
[tex]\[
b = \frac{3}{2}
\][/tex]
3. Square [tex]\( b \)[/tex]:
Now, we square this value to get the constant term required to complete the square. So,
[tex]\[
b^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}
\][/tex]
4. Form the Perfect Square Trinomial:
The left-hand side should match the form [tex]\((x + b)^2\)[/tex]. Hence, the trinomial [tex]\( x^2 + 3x \)[/tex] should be converted to:
[tex]\[
x^2 + 3x + \frac{9}{4}
\][/tex]
5. Identify the Value of [tex]\( c \)[/tex]:
To achieve [tex]\( x^2 + 3x + \frac{9}{4} \)[/tex] on the left-hand side, the value of [tex]\( c \)[/tex] must be:
[tex]\[
c = \frac{9}{4}
\][/tex]
6. Verification:
Substituting [tex]\( c \)[/tex] back into the equation for confirmation:
[tex]\[
x^2 + 3x + \frac{9}{4} = \frac{7}{4} + \frac{9}{4}
\][/tex]
Simplifying the right-hand side:
[tex]\[
\frac{7}{4} + \frac{9}{4} = \frac{16}{4} = 4
\][/tex]
The equation balances.
Thus, the value of [tex]\( c \)[/tex] that makes the equation [tex]\( x^2 + 3x + c \)[/tex] a perfect square trinomial is:
[tex]\[
c = \frac{9}{4} = 2.25
\][/tex]
Additionally, the value of [tex]\( b \)[/tex], which was used to complete the square, is:
[tex]\[
b = \frac{3}{2} = 1.5
\][/tex]
