What number should be added to both sides of the equation to complete the square?
[tex]x^2+3 x=6[/tex]
A. [tex]\frac{3}{2}[/tex]
B. [tex]\left(\frac{3}{2}\right)^2[/tex]
C. 3
D. [tex]6^2[/tex]
Answer (2)
Identify the coefficient of the x term, which is 3.
Calculate ( 2 3 ) 2 .
The result of the calculation is 4 9 .
Therefore, the number to be added to both sides of the equation is ( 2 3 ) 2 .
Explanation
Understanding the problem We are given the equation x 2 + 3 x = 6 and asked to find the number that should be added to both sides 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 to both sides of the equation.
Identifying the coefficient In the given equation, x 2 + 3 x = 6 , the coefficient of the x term is 3 . Therefore, we need to calculate ( 2 3 ) 2 .
Calculating the value Calculating ( 2 3 ) 2 , we get ( 2 3 ) 2 = 2 2 3 2 = 4 9 = 2.25.
Final Answer Therefore, the number that should be added to both sides of the equation to complete the square is 4 9 or 2.25 .
Examples
Completing the square is a useful technique in algebra. For example, suppose you want to find the vertex of the parabola given by the equation y = x 2 + 6 x + 5 . By completing the square, you can rewrite the equation in vertex form, y = ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In this case, you would add and subtract ( 2 6 ) 2 = 9 to the equation to get y = ( x 2 + 6 x + 9 ) − 9 + 5 = ( x + 3 ) 2 − 4 . Thus, the vertex of the parabola is ( − 3 , − 4 ) .
To complete the square for the equation x 2 + 3 x = 6 , we need to add ( 2 3 ) 2 or 4 9 to both sides. Therefore, the chosen option is B. ( 2 3 ) 2 .
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