Okay, so I need to convert the quadratic equation x² + 2.6x + 3.6 into the form (x + b)² + c. Hmm, I remember this is called completing the square. Let me try to recall how that works. 

First, the general form of a quadratic is ax² + bx + c. To complete the square, we take the coefficient of x, divide it by 2, square it, and then add and subtract that value inside the equation. But in this case, the coefficient of x² is 1, which makes things a bit simpler. 

So starting with the given quadratic: x² + 2.6x + 3.6. I need to write this as (x + b)² + c. Let me expand (x + b)² to see what that gives. Expanding it, we get x² + 2bx + b². So comparing this to the original quadratic, which is x² + 2.6x + 3.6, the coefficients of x and the constant term must match. 

Looking at the x terms: in the expanded form, it's 2bx, and in the original quadratic, it's 2.6x. Therefore, 2b must equal 2.6. Solving for b, we divide both sides by 2: b = 2.6 / 2. Let me calculate that. 2.6 divided by 2 is 1.3. So b is 1.3. 

Now, moving on to the constant terms. In the expanded form, the constant term is b², which we have to add, but in the original quadratic, the constant term is 3.6. So we need to adjust for that. Let me write that out. 

Original quadratic: x² + 2.6x + 3.6  
Expanded form: (x + b)² + c = x² + 2bx + b² + c  
So, equating the constants: b² + c = 3.6  
We already found that b = 1.3, so b² is (1.3)². Let me compute that. 1.3 squared is 1.69. 

So substituting back into the equation for the constants: 1.69 + c = 3.6  
Therefore, solving for c, we subtract 1.69 from both sides: c = 3.6 - 1.69. Let me do that subtraction. 3.6 is the same as 3.60. 3.60 - 1.69. Subtracting the hundredths place: 0 - 9, can't do that, borrow from the tenths. So 0 becomes 10, but the tenths place is 6, which becomes 5 after borrowing. Then, 10 - 9 = 1. Now the tenths place: 5 - 6. Again, can't do that, borrow from the units. Units place is 3, becomes 2, tenths place becomes 15. 15 - 6 = 9. Units place: 2 - 1 = 1. So the result is 1.91. Therefore, c = 1.91. 

So now we have b = 1.3 and c = 1.91. The question asks for b + c. Let's add those together. 1.3 + 1.91. Let me add the decimals. 1.3 is the same as 1.30. 1.30 + 1.91. Adding the hundredths: 0 + 1 = 1. Tenths: 3 + 9 = 12. Units: 1 + 1 = 2, plus the carried over 1 makes 3. So the sum is 3.21. 

Wait, let me check that addition again. 1.30 + 1.91. 

Starting from the rightmost digit: 0 + 1 = 1. Next digit: 3 + 9 = 12. Write down 2 and carry over 1. Then the next digit: 1 + 1 (from 1.30) + 1 (carried over) = 3. So yes, 3.21. 

But let me verify if the original quadratic can indeed be written as (x + 1.3)² + 1.91. Let's expand (x + 1.3)² + 1.91. 

First, expand (x + 1.3)²: x² + 2*1.3x + (1.3)² = x² + 2.6x + 1.69. Then add 1.91: x² + 2.6x + 1.69 + 1.91 = x² + 2.6x + (1.69 + 1.91) = x² + 2.6x + 3.6. Which matches the original quadratic. So that checks out. 

Therefore, b is 1.3 and c is 1.91, so their sum is 1.3 + 1.91 = 3.21. 

I think that's the answer. But just to be thorough, let me recap the steps. 

1. Start with quadratic: x² + 2.6x + 3.6.
2. To complete the square, take coefficient of x, which is 2.6, divide by 2 to get 1.3. That's b.
3. Square b to get 1.69. Subtract that from the constant term 3.6 to find c. 3.6 - 1.69 = 1.91.
4. Add b and c: 1.3 + 1.91 = 3.21.
5. Verify by expanding (x + 1.3)² + 1.91, which gives the original quadratic. 

Everything seems to check out. So the answer should be 3.21. 

Wait, but let me make sure there isn't a miscalculation in the subtraction step. 3.6 - 1.69. 

3.6 is 3 and 0.6. 1.69 is 1 and 0.69. So subtracting: 3 - 1 = 2, 0.6 - 0.69. Hmm, 0.6 is less than 0.69, so we need to borrow 1 from the units place. So 3.6 becomes 2.6 + 1.0. Then 2.6 - 1.0 = 1.6, and 1.0 - 0.69 = 0.31. Wait, that can't be. Wait, maybe I should convert to hundredths. 

3.6 is 3.60. 1.69 is 1.69. 3.60 - 1.69. 

Starting from the right: 0 - 9. Can't do that, borrow from the tenths place. The tenths place is 6, borrow 1 to make it 10. So hundredths place becomes 10 - 9 = 1. The tenths place is now 5. Then subtract the tenths place: 5 - 6. Can't do that, borrow from the units place. Units place is 3, becomes 2. Tenths place becomes 15. 15 - 6 = 9. Units place: 2 - 1 = 1. So total is 1.91. Yes, correct. So 3.6 - 1.69 = 1.91. 

Therefore, c is indeed 1.91. 

Another way to check: if I use b = 1.3, then (x + 1.3)^2 = x² + 2.6x + 1.69. Adding c gives x² + 2.6x + 1.69 + c. To make the constant term 3.6, 1.69 + c = 3.6. So c = 3.6 - 1.69 = 1.91. 

All steps are correct. So the answer is 3.21. 

I think that's solid. No mistakes detected in the process. The key was remembering how completing the square works, especially since the coefficient of x² is 1. Then carefully computing b and c, making sure the arithmetic is correct. The verification step by expanding the completed square form confirmed that everything matches the original quadratic, which gives me confidence that the answer is right.
Therefore, the value of $b + c$ is $\boxed{3.21}$.
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