Solve : `sqrt(2x+7)=x+2`

Answer

To solve the equation \( \sqrt{2x + 7} = x + 2 \), we will follow these steps:

### Step 1: Square both sides of the equation
To eliminate the square root, we square both sides:
\[ (\sqrt{2x + 7})^2 = (x + 2)^2 \]
This simplifies to:
\[ 2x + 7 = (x + 2)(x + 2) \]

### Step 2: Expand the right side
Now, we expand the right side using the formula for squaring a binomial:
\[ 2x + 7 = x^2 + 4x + 4 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ 0 = x^2 + 4x + 4 - 2x - 7 \]
This simplifies to:
\[ 0 = x^2 + 2x - 3 \]

### Step 4: Factor the quadratic equation
Now we need to factor the quadratic equation \( x^2 + 2x - 3 \). We look for two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(3\) and \(-1\) fit this requirement:
\[ 0 = (x + 3)(x - 1) \]

### Step 5: Solve for \(x\)
Now we set each factor equal to zero:
1. \(x + 3 = 0\) → \(x = -3\)
2. \(x - 1 = 0\) → \(x = 1\)

### Step 6: Check for extraneous solutions
We need to check if these solutions satisfy the original equation:
1. For \(x = -3\):
\[ \sqrt{2(-3) + 7} = \sqrt{-6 + 7} = \sqrt{1} = 1 \]
and
\[ -3 + 2 = -1 \quad \text{(not a solution)} \]

2. For \(x = 1\):
\[ \sqrt{2(1) + 7} = \sqrt{2 + 7} = \sqrt{9} = 3 \]
and
\[ 1 + 2 = 3 \quad \text{(valid solution)} \]

### Final Solution
Thus, the only valid solution is:
\[ \boxed{1} \]

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