Final Answer:
The completed square and solutions to the equation are:
Completed square: (x + 1/2)² = 0
Solutions: x = -1/2
Step-by-step explanation:
Move constant term to the right side:
1/4x² + x = -1/4
Isolate the quadratic term:
1/4x² = -1/4 - x
Multiply both sides by 4:
x² = -1 - 4x
Complete the square:
Add (1/2)² = 1/4 to both sides, which is half of the coefficient of x². x² + 1/4 = -1 - 4x + 1/4
Rewrite the left side as a squared term: (x + 1/2)² = -4x
Solve for x: Set the squared term equal to 0 and solve for x: (x + 1/2)² = 0 x + 1/2 = 0 x = -1/2
Therefore, the completed square is (x + 1/2)² = 0, and the only solution to the equation is x = -1/2.