Question

Solve the following equation by completing the square.
1/4x^2+x+1/4=0

Answer 1

Since the first term is the square of 1/2x and the second term is twice the product of 1/2x and 1, we want to complete the following square:

`\left((1)/(2)x+1\right)^2=(1)/(4)x^2+x+1`

To do so, we add 3/4 to both sides:

`(1)/(4)x^2+x+(1)/(4)=0 \iff (1)/(4)x^2+x+1=(3)/(4)`

So, we can now write the equation as

`\left((1)/(2)x+1\right)^2=(3)/(4)`

And then continue as usual:

`(1)/(2)x+1\right=\pm(√(3))/(2)`

`(1)/(2)x=\pm(√(3))/(2)-1`

`x=\pm√(3)-2`

Answer 2

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.