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

All the expansions for the expressions provided in the question have been verified to be correct using algebraic methods. Each expansion matches the given form after applying the distributive property or the binomial theorem. Therefore, every expression listed in the question is accurate.
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Answered by Anonymous | 2025-07-04

Expand each given expression using the distributive property or the binomial theorem.
Compare the expanded form with the given right-hand side.
Verify that each expansion matches the given expression.
Conclude that all the given expansions are correct: All expansions are correct. ​

Explanation

Problem Analysis We are given a set of algebraic expressions and their expansions, and our task is to verify the correctness of each expansion. We will expand each expression using the distributive property or the binomial theorem and compare the result with the given expansion.

Verifying the Expansions



( x + 6 ) ( x + 6 ) = x 2 + 6 x + 6 x + 36 = x 2 + 12 x + 36 . The given expansion is x 2 + 12 x + 36 , which matches our result. Thus, the expansion is correct.

( 2 x + 5 ) ( 2 x + 5 ) = ( 2 x ) 2 + 2 ( 2 x ) ( 5 ) + 5 2 = 4 x 2 + 20 x + 25 . The given expansion is 4 x 2 + 20 x + 25 , which matches our result. Thus, the expansion is correct.

( 3 x − 4 ) ( 3 x − 4 ) = ( 3 x ) 2 − 2 ( 3 x ) ( 4 ) + ( − 4 ) 2 = 9 x 2 − 24 x + 16 . The given expansion is 9 x 2 − 24 x + 16 , which matches our result. Thus, the expansion is correct.

( x + 8 ) 2 = x 2 + 2 ( x ) ( 8 ) + 8 2 = x 2 + 16 x + 64 . The given expansion is x 2 + 16 x + 64 , which matches our result. Thus, the expansion is correct.

( 4 x − 3 ) 2 = ( 4 x ) 2 − 2 ( 4 x ) ( 3 ) + ( − 3 ) 2 = 16 x 2 − 24 x + 9 . The given expansion is 16 x 2 − 24 x + 9 , which matches our result. Thus, the expansion is correct.



Conclusion All the given expansions are correct.

Examples
Understanding how to expand algebraic expressions like these is fundamental in many areas, such as physics, engineering, and computer science. For example, when calculating the trajectory of a projectile, you often need to expand expressions involving velocities and time. Similarly, in computer graphics, transformations of objects in 3D space rely on expanding expressions involving matrices and vectors. Mastering these algebraic manipulations provides a solid foundation for tackling more complex problems in various fields.

Answered by GinnyAnswer | 2025-07-04