Find the value that completes the square: [tex]x^2+\frac{1}{2} x+\square=2+ \square[/tex]
Answer (1)
Identify the coefficient of the x term, which is 2 1 .
Divide the coefficient by 2: 2 1 ÷ 2 = 4 1 .
Square the result: ( 4 1 ) 2 = 16 1 .
Add this value to both sides of the equation to complete the square: 16 1 .
Explanation
Understanding the Problem We are given the equation x 2 + 2 1 x + □ = 2 + □ . Our goal is to find the value that completes the square on the left side of the equation. Completing the square means we want to find a value such that the left side can be written in the form ( x + a ) 2 for some a .
Finding the Value to Complete the Square To complete the square for a quadratic expression of the form x 2 + b x , we need to add ( 2 b ) 2 . In our case, b = 2 1 . So, we need to add ( 2 1 ⋅ 2 1 ) 2 to both sides of the equation.
Calculating the Value Let's calculate the value we need to add: ( 2 1 ⋅ 2 1 ) 2 = ( 4 1 ) 2 = 16 1 . Therefore, we need to add 16 1 to both sides of the equation to complete the square.
Completing the Square The equation becomes x 2 + 2 1 x + 16 1 = 2 + 16 1 . The left side is now a perfect square: ( x + 4 1 ) 2 = 2 + 16 1 .
Final Answer Thus, the value to be added to both sides of the equation is 16 1 .
Examples
Completing the square is a useful technique in many areas, such as finding the vertex of a parabola or solving quadratic equations. For example, suppose you are designing a parabolic arch for a bridge. Knowing how to complete the square allows you to determine the maximum height of the arch and where it occurs, ensuring the bridge is structurally sound and aesthetically pleasing. By completing the square, you can rewrite the equation of the arch in vertex form, making it easy to identify the vertex coordinates.
