https://file.aiquickdraw.com/mathbot/file/1726195407014js9hwkkx.jpg

The equation from the image is:

\[
x + 1 - 2\sqrt{x + 4} = 0
\]

To solve this, we can follow these steps:

### Step 1: Isolate the square root term
Move the other terms to the right-hand side of the equation:
\[
2\sqrt{x + 4} = x + 1
\]

### Step 2: Square both sides
Squaring both sides eliminates the square root:
\[
(2\sqrt{x + 4})^2 = (x + 1)^2
\]
\[
4(x + 4) = (x + 1)^2
\]

### Step 3: Expand both sides
Expand both the left and right-hand sides:
\[
4(x + 4) = 4x + 16
\]
On the right-hand side:
\[
(x + 1)^2 = x^2 + 2x + 1
\]

So, the equation becomes:
\[
4x + 16 = x^2 + 2x + 1
\]

### Step 4: Move all terms to one side
Rearrange the equation to set it equal to 0:
\[
0 = x^2 + 2x + 1 - 4x - 16
\]
Simplify:
\[
x^2 - 2x - 15 = 0
\]

### Step 5: Solve the quadratic equation
This is a standard quadratic equation. We can factor it:
\[
(x - 5)(x + 3) = 0
\]

So, the solutions are:
\[
x = 5 \quad \text{or} \quad x = -3
\]

### Step 6: Check for extraneous solutions
Since we squared both sides in the process, we need to check if both solutions are valid by substituting them back into the original equation:

- For \( x = 5 \):
  \[
  5 + 1 - 2\sqrt{5 + 4} = 6 - 2\sqrt{9} = 6 - 6 = 0
  \]
  This is valid.

- For \( x = -3 \):
  \[
  -3 + 1 - 2\sqrt{-3 + 4} = -2 - 2\sqrt{1} = -2 - 2 = -4
  \]
  This does not satisfy the original equation.

Thus, the only valid solution is:
\[
x = 5
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve an equation with multiple square root terms?
2. How does squaring both sides of an equation affect possible solutions?
3. Can a quadratic equation have more than two solutions?
4. What are the conditions for a solution to be extraneous?
5. How can you solve equations with cube roots?

### Tip:
Always check for extraneous solutions when solving equations that involve squaring both sides!

Solve the equation x + 1 - 2√(x + 4) = 0.

Learn how to solve the radical equation x + 1 - 2√(x + 4) = 0 with a step-by-step solution. This problem includes key concepts such as isolating square roots, solving quadratic equations, and checking for extraneous solutions. Suitable for students in Grades 9-11.

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