Let's analyze whether the student’s conclusion that [tex]\( x = 4 \)[/tex] is the solution to the equation [tex]\( \sqrt{2x + 1} + 3 = 0 \)[/tex].
1. Substitute [tex]\( x = 4 \)[/tex] into the equation:
We start by substituting [tex]\( x = 4 \)[/tex] into the given equation.
[tex]\[
\sqrt{2(4) + 1} + 3
\][/tex]
2. Simplify the expression inside the square root:
Calculate inside the square root first:
[tex]\[
2(4) + 1 = 8 + 1 = 9
\][/tex]
3. Take the square root of the result:
Now, find the square root:
[tex]\[
\sqrt{9} = 3
\][/tex]
4. Add 3 to the result of the square root:
Finally, add 3 to the square root value:
[tex]\[
3 + 3 = 6
\][/tex]
So, after substituting [tex]\( x = 4 \)[/tex] into the left side of the equation [tex]\( \sqrt{2x + 1} + 3 \)[/tex], we get a value of 6, not 0.
Therefore, the expression [tex]\( \sqrt{2x + 1} + 3 \)[/tex] simplifies to 6 when [tex]\( x = 4 \)[/tex]. Because 6 does not equal 0, the left side of the equation does not equal the right side when [tex]\( x = 4 \)[/tex].
Conclusion:
The student’s conclusion that [tex]\( x = 4 \)[/tex] is the solution to the equation [tex]\( \sqrt{2x + 1} + 3 = 0 \)[/tex] is incorrect because substituting [tex]\( x = 4 \)[/tex] does not satisfy the equation. The correct approach should lead to a situation where the simplified left side equals zero, but here it equals 6.
