Factor the following: [tex]81 x^2+72 x+16[/tex]
Answer (2)
The quadratic expression 81 x 2 + 72 x + 16 factors to ( 9 x + 4 ) 2 since it is a perfect square trinomial. The values of A and B are determined to be 9 and 4, respectively, and the middle term is verified. This factoring process allows for simpler solutions in algebraic problems.
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Recognize the quadratic expression as a potential perfect square trinomial.
Identify A and B such that A 2 = 81 and B 2 = 16 , giving A = 9 and B = 4 .
Verify that the middle term 72 x matches 2 A B x = 2 ( 9 ) ( 4 ) x .
Express the factored form as ( 9 x + 4 ) 2 .
Explanation
Understanding the Problem We are given the quadratic expression 81 x 2 + 72 x + 16 and asked to factor it.
Recognizing the Pattern We observe that the given expression might be a perfect square trinomial. A perfect square trinomial has the form ( A x + B ) 2 = A 2 x 2 + 2 A B x + B 2 .
Finding A and B In our case, we have A 2 = 81 , which means A = 9 . Also, B 2 = 16 , which means B = 4 . Now we need to check if the middle term 72 x matches 2 A B x .
Verifying the Middle Term Let's calculate 2 A B x = 2 ( 9 ) ( 4 ) x = 72 x . Since the middle term of the given expression is indeed 72 x , we can confirm that the given expression is a perfect square trinomial.
Factoring the Expression Therefore, the factored form of the expression is ( 9 x + 4 ) 2 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where the area is represented by the quadratic expression 81 x 2 + 72 x + 16 . By factoring this expression to ( 9 x + 4 ) 2 , you determine that the garden is a square with each side measuring 9 x + 4 units. This helps in planning the layout and fencing required for the garden.
