If `(a+3)sqrt2 + 3 = b sqrt8 + a - 1` then value of a + b is

Options

A. 3
B. 6
C. `13/2`
D. `15/2`

Answer

To solve the equation \((a + 3)\sqrt{2} + 3 = b\sqrt{8} + a - 1\), we can follow these steps:

### Step 1: Rewrite \(\sqrt{8}\)
We start by rewriting \(\sqrt{8}\) in terms of \(\sqrt{2}\):
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
Thus, we can rewrite the equation as:
\[ (a + 3)\sqrt{2} + 3 = b(2\sqrt{2}) + a - 1 \]
This simplifies to:
\[ (a + 3)\sqrt{2} + 3 = 2b\sqrt{2} + a - 1 \]

### Step 2: Separate the terms with \(\sqrt{2}\) and constant terms
Now, we can separate the terms involving \(\sqrt{2}\) from the constant terms:
\[ (a + 3)\sqrt{2} = 2b\sqrt{2} + (a - 1 - 3) \]
This gives us:
\[ (a + 3)\sqrt{2} = 2b\sqrt{2} + (a - 4) \]

### Step 3: Equate the coefficients of \(\sqrt{2}\)
Now, we can equate the coefficients of \(\sqrt{2}\) from both sides:
\[ a + 3 = 2b \]

### Step 4: Equate the constant terms
Next, we equate the constant terms:
\[ 3 = a - 4 \]

### Step 5: Solve for \(a\)
From the equation \(3 = a - 4\), we can solve for \(a\):
\[ a = 3 + 4 = 7 \]

### Step 6: Substitute \(a\) back to find \(b\)
Now, substitute \(a = 7\) back into the equation \(a + 3 = 2b\):
\[ 7 + 3 = 2b \implies 10 = 2b \implies b = \frac{10}{2} = 5 \]

### Step 7: Find \(a + b\)
Finally, we can find \(a + b\):
\[ a + b = 7 + 5 = 12 \]

Thus, the value of \(a + b\) is \(12\).