Answer :

GinnyAnswer
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]

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