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Math  /  Algebra

QuestionFactor completely. 16a2−8116 a^{2}-81

Studdy Solution

STEP 1

1. The expression 16a2−8116a^2 - 81 is a difference of squares.
2. The difference of squares formula can be applied: x2−y2=(x−y)(x+y)x^2 - y^2 = (x-y)(x+y).

STEP 2

1. Identify the terms as perfect squares.
2. Apply the difference of squares formula.
3. Verify the factorization.

STEP 3

Identify the terms as perfect squares:
16a2=(4a)2 16a^2 = (4a)^2 81=92 81 = 9^2
Thus, the expression 16a2−8116a^2 - 81 is a difference of squares.

STEP 4

Apply the difference of squares formula:
16a2−81=(4a)2−92=(4a−9)(4a+9) 16a^2 - 81 = (4a)^2 - 9^2 = (4a - 9)(4a + 9)

STEP 5

Verify the factorization by expanding:
(4a−9)(4a+9)=4a×4a+4a×9−9×4a−9×9 (4a - 9)(4a + 9) = 4a \times 4a + 4a \times 9 - 9 \times 4a - 9 \times 9 =16a2+36a−36a−81 = 16a^2 + 36a - 36a - 81 =16a2−81 = 16a^2 - 81
The factorization is verified as correct.
The completely factored form of the expression is:
(4a−9)(4a+9) \boxed{(4a - 9)(4a + 9)}

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