What number should be added to both sides of the equation to complete the square?

[tex]x^2-6 x=5[/tex]

To complete the square for x 2 − 6 x = 5 , we need to add ( 2 b ​ ) 2 to both sides, where b = − 6 .
Calculate ( 2 − 6 ​ ) 2 = ( − 3 ) 2 = 9 .
Adding 9 to both sides completes the square: x 2 − 6 x + 9 = ( x − 3 ) 2 .
The number to add is 9 ​ .

Explanation

Understanding the problem We are given the equation x 2 − 6 x = 5 and we want to find the number that should be added to both sides to complete the square.

Completing 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. In our case, b = − 6 .

Calculation We calculate ( 2 b ​ ) 2 = ( 2 − 6 ​ ) 2 = ( − 3 ) 2 = 9 . Therefore, the number that should be added to both sides of the equation to complete the square is 9.

Verification Adding 9 to both sides of the original equation, we get: x 2 − 6 x + 9 = 5 + 9 ( x − 3 ) 2 = 14


Examples
Completing the square is a useful technique in algebra. For example, suppose you want to find the vertex of the parabola defined by the equation y = x 2 − 6 x + 5 . By completing the square, you can rewrite the equation as y = ( x − 3 ) 2 − 4 . From this form, it is easy to see that the vertex of the parabola is at the point ( 3 , − 4 ) . This technique is also used in calculus to evaluate certain integrals and in physics to solve problems involving harmonic oscillators.

Answered by GinnyAnswer | 2025-07-03

To complete the square for the equation x 2 − 6 x = 5 , you need to add 9 to both sides. This results in the equation transforming into ( x − 3 ) 2 = 14 . Thus, the number to be added is 9 .
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Answered by Anonymous | 2025-07-04