To solve the equation [tex]\(\sqrt{6 - 2w} = w - 1\)[/tex], follow these steps:
1. Isolate the square root:
The equation is already isolated on one side.
[tex]\[
\sqrt{6 - 2w} = w - 1
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{6 - 2w})^2 = (w - 1)^2
\][/tex]
This simplifies to:
[tex]\[
6 - 2w = (w - 1)^2
\][/tex]
3. Expand and simplify the right side of the equation:
[tex]\[
6 - 2w = w^2 - 2w + 1
\][/tex]
4. Rearrange the equation to set it to zero and form a quadratic equation:
[tex]\[
w^2 - 2w + 1 - 6 + 2w = 0
\][/tex]
Simplify by combining like terms:
[tex]\[
w^2 - 5 = 0
\][/tex]
5. Solve the quadratic equation:
[tex]\[
w^2 = 5
\][/tex]
Take the square root of both sides:
[tex]\[
w = \pm\sqrt{5}
\][/tex]
6. Verify the solutions in the original equation to check for extraneous roots:
- For [tex]\( w = \sqrt{5} \)[/tex]:
[tex]\[
\sqrt{6 - 2\sqrt{5}} = \sqrt{5} - 1
\][/tex]
Substituting [tex]\( \sqrt{5} \)[/tex] in the original equation:
[tex]\[
\sqrt{6 - 2\sqrt{5}} = \sqrt{5} - 1
\][/tex]
Calculate the left-hand side:
[tex]\[
\sqrt{6 - 2\sqrt{5}} = \sqrt{6 - 2(\sqrt{5})} = \sqrt{6 - 2\sqrt{5}}
\][/tex]
Observe that [tex]\( \sqrt{6 - 2\sqrt{5}} \neq \sqrt{5} - 1 \)[/tex], hence it doesn't work.
- For [tex]\( w = -\sqrt{5} \)[/tex]:
[tex]\[
\sqrt{6 - 2(-\sqrt{5})} = -\sqrt{5} - 1
\][/tex]
The left-hand side simplifies to:
[tex]\[
\sqrt{6 + 2\sqrt{5}}
\][/tex]
The right-hand side:
[tex]\[
-\sqrt{5} - 1
\][/tex]
Once again, these do not match.
Thus, considering the solution verification step, one root is valid:
[tex]\[
w = \sqrt{5}
\][/tex]
