Question
Factor the following. \[ (y-6)^{2}-81 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( (y-6)^{2}-81=\square \) (Simplify your answer. Factor completely.) B. \( (y-6)^{2}-81 \) is prime.
Ask by Weaver Olson. in the United States
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (y - 6)^2 - 81 = (y - 15)(y + 3) \)
Solution
The given expression is (y - 6)² - 81. Notice that 81 is 9², so we have a difference of two squares:
(y - 6)² - 9²
Using the difference of squares formula, a² - b² = (a - b)(a + b), let a = y - 6 and b = 9. Then the expression factors as:
(y - 6 - 9)(y - 6 + 9) = (y - 15)(y + 3)
Thus, the correct choice is:
A. (y - 6)² - 81 = (y - 15)(y + 3)
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Simplify this solution Mind Expander
To factor the expression \((y-6)^{2}-81\), we recognize it as a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a = (y-6)\) and \(b = 9\) (since \(81\) is \(9^2\)). Applying the formula, we have: \[ (y-6)^{2} - 9^{2} = \left((y-6) - 9\right)\left((y-6) + 9\right) \] This simplifies to: \[ (y - 15)(y + 3) \] So, the completely factored form is \((y - 15)(y + 3)\). Hence, the correct choice is: A. \((y-6)^{2}-81=(y-15)(y+3)\) (Simplify your answer. Factor completely.)
