What number should be added to both sides of the equation to complete the square? [tex]x^2+12 x=11[/tex]
Answer (1)
Identify the coefficient of the x term: b = 12 .
Calculate the value to add to both sides: ( 2 b ) 2 = ( 2 12 ) 2 = 6 2 = 36 .
Add 36 to both sides of the equation to complete the square.
The number to be added is 36 .
Explanation
Understanding the Problem We are given the equation x 2 + 12 x = 11 and we want to find the number that should be added to both sides to complete the square. Completing the square is a technique used to rewrite a quadratic expression in the form ( x + a ) 2 + b , which makes it easier to solve the equation or analyze the function.
Identifying the Coefficient To complete the square for a quadratic expression of the form x 2 + b x , we need to add ( 2 b ) 2 to both sides of the equation. This is because ( x + 2 b ) 2 = x 2 + b x + ( 2 b ) 2 . In our equation, x 2 + 12 x = 11 , we can see that b = 12 .
Calculating the Value Now, we need to calculate ( 2 12 ) 2 . First, divide 12 by 2: 2 12 = 6 . Then, square the result: 6 2 = 36 . Therefore, we need to add 36 to both sides of the equation to complete the square.
Verifying the Solution Adding 36 to both sides of the original equation, we get: x 2 + 12 x + 36 = 11 + 36 , which simplifies to ( x + 6 ) 2 = 47 . This confirms that adding 36 completes the square.
Final Answer The number that should be added to both sides of the equation to complete the square is 36.
Examples
Completing the square is a useful technique in various real-world applications. For example, consider a farmer who wants to enclose a rectangular area for his livestock using a fixed amount of fencing. By completing the square, he can determine the dimensions of the rectangle that maximize the enclosed area. This optimization problem demonstrates the practical value of completing the square in resource management and planning.
