Answer:
A
Step-by-step explanation:
(2x^2 + 3x - 4) (x + 4)
FOIL.
2x^2 times x = 2x^3
3x times x is 3x^2
-4 times x = -4x
2x^2 times 4 = 8x^2
3x times 4 = 12 x
-4 times 4 = -16
Combine like terms to get 2x^3 + 11x^2 + 8x - 16.
Answer:
14). 2nd quadrant
15). 1st quadrant
Step-by-step explanation:
14).Coordinates of a point → J(-8, -12)
Coordinates of the new point J' after reflection of x-axis will follow the rule,
(x, y) → (x, -y)
Coordinates of J' → (-8, 12)
Therefore, point J' will lie in 2nd quadrant.
15). Coordinates of a point → W(-6, 7)
Rule for the rotation by 90°clockwise about the origin,
(x, y) → (y, -x)
Coordinates of point W → (-6, 7)
Following this rule,
W(-6, 7) → W'(7, 6)
Therefore, point W' will lie in the first quadrant.
Step-by-step explanation:
step 1. Where is the original equation? let's assume it is y = x^2
step 2. to shift to the right 3 places replace x with x - 3 and to shift 4 down subtract 4 from the equation
step 3. y = (x - 3)^2 - 4.
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 ; <u>AND</u>: " (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 ;
{<u>Note</u>: 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 = </span>(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 "<u>difference of squares</u>".}.
→ 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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