BRAINLIESSTTTT ASAP !!!!!!!!!! 20 pointssss

1 answer:

7 0

Answers:
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Part A) " (3x + 4) " units .
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Part B) "The dimensions of the rectangle are:

" (4x + 5y) " units ; AND: " (4x − 5y)" units."
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Explanation for Part A):
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Since each side length of a square is the same;

Area = Length * width = L * w ; L = w = s = s ;

in which: " s = side length" ;

So, the Area of a square, "A" = L * w = s * s = s² ;

{Note: A "square" is a rectangle with 4 (four) equal sides.}.

→ Each side length, "s", of a square is equal.

Given: s² = "(9x² + 24x + 16)" square units ;

Find "s" by factoring: "(9x² + 24x + 16)" completely:

→ " 9x² + 24x + 16 ";

Factor by "breaking into groups" :

"(9x² + 24x + 16)" =

 → "(9x² + 12x) (12x + 16)" ;
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Given: " (9x² + 24x + 16) " ;
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Let us start with the term:
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" (9x² + 12x) " ;

 → Factor out a "3x" ; → as follows:
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 → " 3x (3x + 4) " ;

Then, take the term:
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→ " (12x + 16) " ;

And factor out a "4" ; → as follows:
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→ " 4 (3x + 4) "
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We have:

" 9x² + 24x + 16 " ;

= " 3x (3x + 4) + 4(3x + 4) " ;
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Now, notice the term: "(3x + 4)" ;

We can "factor out" this term:

3x (3x + 4) + 4(3x + 4) =

→ " (3x + 4) (3x + 4) " . → which is the fully factored form of:

" 9x² + 24x + 16 " ;
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→ Or; write: " (3x + 4) (3x + 4)" ; as: " (3x + 4)² " .
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→ So, "s² = 9x² + 24x + 16 " ;

Rewrite as: " s² = (3x + 4)² " .

→ Solve for the "positive value of "s" ;

→ {since the "side length of a square" cannot be a "negative" value.}.
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→ Take the "positive square root of EACH SIDE of the equation;
to isolate "s" on one side of the equation; & to solve for "s" ;

→ ⁺√(s²) = ⁺√[(3x + 4)²] '

To get:

→ s = " (3x + 4)" units .
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Part A): The answer is: "(3x + 4)" units.
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Explanation for Part B):
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The area, "A" of a rectangle is:

A = L * w ;

in which "A" is the "area" of the rectangle;
"L" is the "length" of the rectangle;
"w" is the "width" of the rectangle;
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Given: " A = (16x² − 25y²) square units" ;

→ We are asked to find the dimensions, "L" & "w" ;
→ by factoring the given "area" expression completely:
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→ Factor: " (16x² − 25y²) square units " completely '

Note that: "16" and: "25" are both "perfect squares" ;

We can rewrite: " (16x² − 25y²) " ; as:

= " (4²x²) − (5²y²) " ; and further rewrite the expression:
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Note:
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" (16x²) " ; can be written as: "(4x)² " ;

↔ " (4x)² = "(4²)(x²)" = 16x² "

Note: The following property of exponents:

→ (xy)ⁿ = xⁿ yⁿ ; → As such: " 16x² = (4²x²) = (4x)² " .
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Note:
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→ " (25x²) " ; can be written as: " (5x)² " ;

↔ "( 5x)² = "(5²)(x²)" = 25x² " ;

Note: The following property of exponents:

→ (xy)ⁿ = xⁿ yⁿ ; → As such: " 25x² = (5²x²) = (5x)² " .
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→ So, we can rewrite: " (16x² − 25y²) " ;

as: " (4x)² − (5y)² " ;

→ {Note: We substitute: "(4x)² " for "(16x²)" ; & "(5y)² " for "(25y²)" .} . ;
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→ We have: " (4x)² − (5y)² " ;

→ Note that we are asked to "factor completely" ;

→ Note that: " x² − y² = (x + y) (x − y) " ;

→ {This property is known as the "difference of squares".}.

→ As such: " (4x)² − (5y)² " = " (4x + 5y) (4x − 5y) " .
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Part B): The answer is: "The dimensions of the rectangle are:

" (4x + 5y) " units ; AND: " (4x − 5y)" units."
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