$x^2+\frac{1}{2} x+$
Answer (1)
To complete the square for the expression x 2 + 2 1 x + , we need to find the constant term that makes it a perfect square trinomial.
We use the formula c = ( 2 b ) 2 , where b = 2 1 .
Calculating c , we get c = ( 2 1/2 ) 2 = 16 1 .
Therefore, the missing term is 16 1 .
Explanation
Problem Analysis The given expression x 2 + 2 1 x + is incomplete. It seems like the intention was to complete the square, but the constant term is missing. To complete the square, we need to add a constant term such that the expression becomes a perfect square trinomial.
Completing the Square To complete the square for a quadratic expression of the form x 2 + b x + c , we need to find a value for c such that c = ( 2 b ) 2 . In this case, b = 2 1 , so we need to find c such that c = ( 2 1/2 ) 2 = ( 4 1 ) 2 = 16 1 . Therefore, the complete expression would be x 2 + 2 1 x + 16 1 .
Perfect Square Trinomial The completed expression x 2 + 2 1 x + 16 1 can be written as a perfect square: ( x + 4 1 ) 2 . Thus, the missing term is 16 1 .
Final Answer The missing term to complete the square is 16 1 .
Examples
Completing the square is a useful technique in algebra. For example, it can be used to rewrite the equation of a circle in standard form, making it easier to identify the center and radius of the circle. It is also used in calculus to evaluate certain integrals and to solve optimization problems.
