Amira is solving the equation [tex]$x^2-6 x=1$[/tex]. Which value must be added to both sides of the equation to make the left side a perfect-square trinomial?
A. -9
B. 8
C. 9
D. 36
Answer (2)
Identify the coefficient of the x term, which is -6.
Divide the coefficient by 2: 2 − 6 = − 3 .
Square the result: ( − 3 ) 2 = 9 .
Add 9 to both sides of the equation to complete the square: 9 .
Explanation
Understanding the Problem We are given the equation x 2 − 6 x = 1 and we want to find the value that must be added to both sides to make the left side a perfect-square trinomial. A perfect-square trinomial is a trinomial that can be factored into the square of a binomial. In general, x 2 + b x + c can be made into a perfect square by adding ( 2 b ) 2 to it.
Finding the Value to Add In our case, we have x 2 − 6 x . Here, b = − 6 . To complete the square, we need to add ( 2 − 6 ) 2 to the left side of the equation. Let's calculate this value.
Calculating the Value We have ( 2 − 6 ) 2 = ( − 3 ) 2 = 9 . Therefore, we must add 9 to both sides of the equation to make the left side a perfect-square trinomial.
Verifying the Result Adding 9 to both sides of the equation x 2 − 6 x = 1 , we get x 2 − 6 x + 9 = 1 + 9 , which simplifies to ( x − 3 ) 2 = 10 . The left side is now a perfect square.
Final Answer Thus, the value that must be added to both sides of the equation to make the left side a perfect-square trinomial is 9.
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, which makes it easy to identify the center and radius of the circle. Suppose you have the equation x 2 + y 2 − 4 x + 6 y − 3 = 0 . By completing the square for both the x and y terms, you can rewrite the equation as ( x − 2 ) 2 + ( y + 3 ) 2 = 16 , which tells you that the circle has center ( 2 , − 3 ) and radius 4. This technique is also used in calculus to evaluate certain integrals and solve differential equations.
To convert the left side of the equation x 2 − 6 x = 1 into a perfect-square trinomial, we must add 9 to both sides. The answer is option C, which is 9.
;
