Select the correct answer.
Which equation is equivalent to the given equation?
[tex]$x^2-6 x=8$[/tex]
A. [tex]$(x-3)^2=17$[/tex]
B. [tex]$(x-6)^2=20$[/tex]
C. [tex]$(x-6)^2=44$[/tex]
D. [tex]$(x-3)^2=14$[/tex]
Answer (1)
Take half of the coefficient of the x term: 2 − 6 = − 3 .
Square the result: ( − 3 ) 2 = 9 .
Add 9 to both sides of the equation: x 2 − 6 x + 9 = 8 + 9 .
Rewrite the left side as a squared term and simplify the right side: ( x − 3 ) 2 = 17 . The answer is ( x − 3 ) 2 = 17 .
Explanation
Understanding the Problem We are given the equation x 2 − 6 x = 8 and asked to find an equivalent equation among the given choices. The given choices are in the form of a completed square, so we will complete the square for the given equation.
Finding the Constant Term To complete the square, we take half of the coefficient of the x term, which is − 6 . Half of − 6 is − 3 .
Adding to Both Sides We add ( − 3 ) 2 = 9 to both sides of the equation to complete the square: x 2 − 6 x + 9 = 8 + 9 .
Rewriting as a Square We rewrite the left side as a squared term: ( x − 3 ) 2 = 17 .
Selecting the Correct Answer Comparing the resulting equation ( x − 3 ) 2 = 17 with the given choices, we see that it matches option A.
Examples
Completing the square is a useful technique in algebra. For example, if you are trying to find the vertex of a parabola given by the equation y = x 2 − 6 x + 8 , you can complete the square to rewrite the equation in vertex form, y = ( x − 3 ) 2 − 1 . This tells you that the vertex of the parabola is at the point ( 3 , − 1 ) . Completing the square can also be used to solve quadratic equations and to put equations of circles and ellipses into standard form.
