$\begin{array}{l} \rightarrow (-3)^2=9 \\ 2 x)^2=4 x^2(-3)^2=9 \\ (2 x-\underbrace{3)(2 x-3)}_{2(-3)(2 x)}=4 x^2-12 x+9 \end{array}$ 1) $(x+6)(x+6) \quad x^2+12 x+36$ 2) $(2 x+5)(2 x+5)$ $\qquad$ $4 x^2+20 x+25$ 3) $(3 x-4)(3 x-4) 9 x^2-24 x+16$ 4) $(x+8)^2 \quad x^2+16 x+64$ 5) $(4 x-3)^2 \quad 16 x^2-24 x+9$

Each given expansion of the squared binomials was verified and found to be accurate. The expansions used the appropriate formulas, and the results matched the provided expressions. Therefore, all expansions are confirmed to be correct.
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Answered by Anonymous | 2025-07-04

Expand each squared binomial using the formulas ( a + b ) 2 = a 2 + 2 ab + b 2 and ( a − b ) 2 = a 2 − 2 ab + b 2 .
Compare each expansion with the given result.
Verify that all the given expansions are correct.
Conclude that all expansions are correct.

Explanation

Problem Analysis We are given a series of expansions of squared binomials and asked to verify their correctness. We will expand each binomial and compare it to the given result.

Expansion 1



( x + 6 ) ( x + 6 ) = ( x + 6 ) 2 . Using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , we have ( x + 6 ) 2 = x 2 + 2 ( x ) ( 6 ) + 6 2 = x 2 + 12 x + 36 . This matches the given expansion x 2 + 12 x + 36 .


Expansion 2


( 2 x + 5 ) ( 2 x + 5 ) = ( 2 x + 5 ) 2 . Using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , we have ( 2 x + 5 ) 2 = ( 2 x ) 2 + 2 ( 2 x ) ( 5 ) + 5 2 = 4 x 2 + 20 x + 25 . This matches the given expansion 4 x 2 + 20 x + 25 .


Expansion 3


( 3 x − 4 ) ( 3 x − 4 ) = ( 3 x − 4 ) 2 . Using the formula ( a − b ) 2 = a 2 − 2 ab + b 2 , we have ( 3 x − 4 ) 2 = ( 3 x ) 2 − 2 ( 3 x ) ( 4 ) + 4 2 = 9 x 2 − 24 x + 16 . This matches the given expansion 9 x 2 − 24 x + 16 .


Expansion 4


( x + 8 ) 2 . Using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , we have ( x + 8 ) 2 = x 2 + 2 ( x ) ( 8 ) + 8 2 = x 2 + 16 x + 64 . This matches the given expansion x 2 + 16 x + 64 .


Expansion 5


( 4 x − 3 ) 2 . Using the formula ( a − b ) 2 = a 2 − 2 ab + b 2 , we have ( 4 x − 3 ) 2 = ( 4 x ) 2 − 2 ( 4 x ) ( 3 ) + 3 2 = 16 x 2 − 24 x + 9 . This matches the given expansion 16 x 2 − 24 x + 9 .


Conclusion All the given expansions are correct.

Examples
Expanding binomials is a fundamental skill in algebra and is used in many real-world applications. For example, when calculating the area of a square garden with sides of length ( x + 5 ) , you would need to expand ( x + 5 ) 2 to find the area in terms of x . This skill is also crucial in physics, engineering, and computer science for modeling various phenomena.

Answered by GinnyAnswer | 2025-07-04