Select the expression that is equivalent to $(x-3)^2$.
A. $x^2-3 x+9$
B. $x^2-6 x+6$
C. $x^2-6 x+9$
D. $x^2-3 x+6$
Answer (1)
Expand the expression ( x − 3 ) 2 using the formula ( a − b ) 2 = a 2 − 2 ab + b 2 .
Substitute a = x and b = 3 into the formula: ( x − 3 ) 2 = x 2 − 2 ( x ) ( 3 ) + 3 2 .
Simplify the expression: ( x − 3 ) 2 = x 2 − 6 x + 9 .
The equivalent expression is x 2 − 6 x + 9 .
Explanation
Understanding the Problem We are given the expression ( x − 3 ) 2 and asked to find an equivalent expression from the given options. The options are: A. x 2 − 3 x + 9 B. x 2 − 6 x + 6 C. x 2 − 6 x + 9 D. x 2 − 3 x + 6
Expanding the Expression To find the equivalent expression, we need to expand ( x − 3 ) 2 . We can use the formula ( a − b ) 2 = a 2 − 2 ab + b 2 . In this case, a = x and b = 3 .
Calculation Applying the formula, we get: ( x − 3 ) 2 = x 2 − 2 ( x ) ( 3 ) + 3 2 = x 2 − 6 x + 9
Comparing with Options Now, we compare the expanded expression x 2 − 6 x + 9 with the given options: A. x 2 − 3 x + 9 B. x 2 − 6 x + 6 C. x 2 − 6 x + 9 D. x 2 − 3 x + 6 The expression x 2 − 6 x + 9 matches option C.
Final Answer Therefore, the expression equivalent to ( x − 3 ) 2 is x 2 − 6 x + 9 .
Examples
Understanding how to expand squared expressions like ( x − 3 ) 2 is useful in many areas of math and science. For example, when solving quadratic equations, completing the square often involves expanding such expressions. In physics, you might encounter similar expansions when dealing with projectile motion or energy calculations. Knowing how to quickly and accurately expand these expressions can save time and prevent errors in more complex problems.
