Answer :
We start with the equation
[tex]$$
\sqrt{x+18} = x-2.
$$[/tex]
Step 1. Square both sides to remove the square root:
[tex]$$
(\sqrt{x+18})^2 = (x-2)^2 \quad \Longrightarrow \quad x+18 = (x-2)^2.
$$[/tex]
Step 2. Expand the right-hand side:
[tex]$$
x+18 = x^2 - 4x + 4.
$$[/tex]
Step 3. Rearrange the equation into standard quadratic form by bringing all terms to one side:
[tex]$$
0 = x^2 - 4x + 4 - x - 18,
$$[/tex]
which simplifies to
[tex]$$
0 = x^2 - 5x - 14.
$$[/tex]
Step 4. Factor the quadratic equation:
[tex]$$
x^2 - 5x - 14 = (x-7)(x+2) = 0.
$$[/tex]
Thus, the possible solutions are
[tex]$$
x = 7 \quad \text{or} \quad x = -2.
$$[/tex]
Step 5. Check each solution in the original equation to ensure they are valid.
For [tex]\( x = 7 \)[/tex]:
- Left-hand side:
[tex]$$
\sqrt{7+18} = \sqrt{25} = 5.
$$[/tex]
- Right-hand side:
[tex]$$
7-2 = 5.
$$[/tex]
Since both sides are equal, [tex]\( x = 7 \)[/tex] is valid.
For [tex]\( x = -2 \)[/tex]:
- Left-hand side:
[tex]$$
\sqrt{-2+18} = \sqrt{16} = 4.
$$[/tex]
- Right-hand side:
[tex]$$
-2-2 = -4.
$$[/tex]
Since [tex]\( 4 \neq -4 \)[/tex], [tex]\( x = -2 \)[/tex] is extraneous.
Thus, the valid solution is:
[tex]$$
x = 7.
$$[/tex]
Among the given choices [tex]\( x=16 \)[/tex], [tex]\( x=9 \)[/tex], [tex]\( x=20 \)[/tex], and [tex]\( x=7 \)[/tex], the correct answer is [tex]\( x=7 \)[/tex].
[tex]$$
\sqrt{x+18} = x-2.
$$[/tex]
Step 1. Square both sides to remove the square root:
[tex]$$
(\sqrt{x+18})^2 = (x-2)^2 \quad \Longrightarrow \quad x+18 = (x-2)^2.
$$[/tex]
Step 2. Expand the right-hand side:
[tex]$$
x+18 = x^2 - 4x + 4.
$$[/tex]
Step 3. Rearrange the equation into standard quadratic form by bringing all terms to one side:
[tex]$$
0 = x^2 - 4x + 4 - x - 18,
$$[/tex]
which simplifies to
[tex]$$
0 = x^2 - 5x - 14.
$$[/tex]
Step 4. Factor the quadratic equation:
[tex]$$
x^2 - 5x - 14 = (x-7)(x+2) = 0.
$$[/tex]
Thus, the possible solutions are
[tex]$$
x = 7 \quad \text{or} \quad x = -2.
$$[/tex]
Step 5. Check each solution in the original equation to ensure they are valid.
For [tex]\( x = 7 \)[/tex]:
- Left-hand side:
[tex]$$
\sqrt{7+18} = \sqrt{25} = 5.
$$[/tex]
- Right-hand side:
[tex]$$
7-2 = 5.
$$[/tex]
Since both sides are equal, [tex]\( x = 7 \)[/tex] is valid.
For [tex]\( x = -2 \)[/tex]:
- Left-hand side:
[tex]$$
\sqrt{-2+18} = \sqrt{16} = 4.
$$[/tex]
- Right-hand side:
[tex]$$
-2-2 = -4.
$$[/tex]
Since [tex]\( 4 \neq -4 \)[/tex], [tex]\( x = -2 \)[/tex] is extraneous.
Thus, the valid solution is:
[tex]$$
x = 7.
$$[/tex]
Among the given choices [tex]\( x=16 \)[/tex], [tex]\( x=9 \)[/tex], [tex]\( x=20 \)[/tex], and [tex]\( x=7 \)[/tex], the correct answer is [tex]\( x=7 \)[/tex].
