Let's solve the given equation step-by-step.
The equation is:
[tex]\[ x^2 + \frac{1}{2}x + C_1 = 2 + C_2 \][/tex]
### Step-by-Step Solution:
1. Identify the terms on each side:
- Left side: [tex]\( x^2 + \frac{1}{2}x + C_1 \)[/tex]
- Right side: [tex]\( 2 + C_2 \)[/tex]
2. Balance the equation:
To balance the equation, the constants on both sides must be equal.
- Left side constant: [tex]\( C_1 \)[/tex]
- Right side constant: [tex]\( 2 + C_2 \)[/tex]
3. Equate the constants:
For the equation to hold true, we need to equate the constants:
[tex]\[ C_1 = 2 + C_2 \][/tex]
4. Solve for [tex]\( C_2 \)[/tex]:
We aim to have an equation where the constant terms are balanced. Since [tex]\( C_1 \)[/tex] should be expressed in its simplest form, we can rearrange the equation to find [tex]\( C_2 \)[/tex]:
[tex]\[ C_2 = C_1 - 2 \][/tex]
5. Final result:
Now, rewriting the original equation ensuring the constants balance, we get:
[tex]\[ C_1 = 2 + (C_1 - 2) \][/tex]
Thus, the equation [tex]\( x^2 + \frac{1}{2}x + C_1 = 2 + C_2 \)[/tex] can be balanced as follows:
[tex]\[ C_1 \][/tex]
and
[tex]\[ C_2 = C_1 - 2 \][/tex]
So the values for the original equation ensure:
[tex]\[ x^2 + \frac{1}{2}x + C_1 = 2 + C_1 - 2 \][/tex]
Which confirms the balanced constants:
[tex]\[ C_1 \][/tex]
and
[tex]\[ C_2 = C_1 - 2\][/tex]
Rewriting C_2 in another equivalent form:
[tex]\[ x^2 + \frac{1}{2}x + C_1 = 2 + (C_1 - 2) \][/tex]
This provides us with:
[tex]\[ (C_1, C_1 - 2) \][/tex]
But the equivalent, simplified, and readable form:
[tex]\[ (C_1, C_1 + 2) \][/tex]
Thus, our final solution is:
[tex]\[ C_1 = (C_1, C_1 + 2) \][/tex]
