Question

for each rectangle with area given, determine the binomial factors that describe the dimensions. area = 6x² + 17x − 3 = blank (rectangle a image) area = 21x² − 13x + 2 = blank (rectangle b image) if x = 3, then which rectangle has the greater area? blank. since rectangle a’s area is blank sq. units and rectangle b’s area is blank sq. units. therefore, rectangle blank area is larger than rectangle blank area.

Explanation:

Part 1: Factor the area of Rectangle A

The area of Rectangle A is given by the quadratic expression \( 6x^2 + 17x - 3 \). We factor this quadratic:

Step 1: Find two numbers that multiply to \( 6 \times (-3) = -18 \) and add up to \( 17 \).

The numbers are \( 18 \) and \( -1 \) (since \( 18 \times (-1) = -18 \) and \( 18 + (-1) = 17 \)).

Step 2: Rewrite the middle term using these numbers.

\( 6x^2 + 18x - x - 3 \)

Step 3: Group and factor.

\( (6x^2 + 18x) - (x + 3) = 6x(x + 3) - 1(x + 3) = (6x - 1)(x + 3) \)

Part 2: Factor the area of Rectangle B

The area of Rectangle B is given by the quadratic expression \( 21x^2 - 13x + 2 \). We factor this quadratic:

Step 1: Find two numbers that multiply to \( 21 \times 2 = 42 \) and add up to \( -13 \).

The numbers are \( -6 \) and \( -7 \) (since \( (-6) \times (-7) = 42 \) and \( -6 + (-7) = -13 \)).

Step 2: Rewrite the middle term using these numbers.

\( 21x^2 - 6x - 7x + 2 \)

Step 3: Group and factor.

\( (21x^2 - 6x) - (7x - 2) = 3x(7x - 2) - 1(7x - 2) = (3x - 1)(7x - 2) \)

Part 3: Calculate the area when \( x = 3 \)

For Rectangle A:

Substitute \( x = 3 \) into \( 6x^2 + 17x - 3 \):
\( 6(3)^2 + 17(3) - 3 = 6(9) + 51 - 3 = 54 + 51 - 3 = 102 \)

For Rectangle B:

Substitute \( x = 3 \) into \( 21x^2 - 13x + 2 \):
\( 21(3)^2 - 13(3) + 2 = 21(9) - 39 + 2 = 189 - 39 + 2 = 152 \)

Part 4: Compare the areas

Rectangle A’s area: \( 102 \) square units
Rectangle B’s area: \( 152 \) square units

Since \( 152 > 102 \), Rectangle B has a larger area.

Final Answers

• Factored form of Rectangle A: \( \boldsymbol{(6x - 1)(x + 3)} \)
• Factored form of Rectangle B: \( \boldsymbol{(3x - 1)(7x - 2)} \)
• Area of A when \( x = 3 \): \( \boldsymbol{102} \)
• Area of B when \( x = 3 \): \( \boldsymbol{152} \)
• Larger area: Rectangle \( \boldsymbol{B} \)

Answer:

Part 1: Factor the area of Rectangle A

The area of Rectangle A is given by the quadratic expression \( 6x^2 + 17x - 3 \). We factor this quadratic:

Step 1: Find two numbers that multiply to \( 6 \times (-3) = -18 \) and add up to \( 17 \).

The numbers are \( 18 \) and \( -1 \) (since \( 18 \times (-1) = -18 \) and \( 18 + (-1) = 17 \)).

Step 2: Rewrite the middle term using these numbers.

\( 6x^2 + 18x - x - 3 \)

Step 3: Group and factor.

\( (6x^2 + 18x) - (x + 3) = 6x(x + 3) - 1(x + 3) = (6x - 1)(x + 3) \)

Part 2: Factor the area of Rectangle B

The area of Rectangle B is given by the quadratic expression \( 21x^2 - 13x + 2 \). We factor this quadratic:

Step 1: Find two numbers that multiply to \( 21 \times 2 = 42 \) and add up to \( -13 \).

The numbers are \( -6 \) and \( -7 \) (since \( (-6) \times (-7) = 42 \) and \( -6 + (-7) = -13 \)).

Step 2: Rewrite the middle term using these numbers.

\( 21x^2 - 6x - 7x + 2 \)

Step 3: Group and factor.

\( (21x^2 - 6x) - (7x - 2) = 3x(7x - 2) - 1(7x - 2) = (3x - 1)(7x - 2) \)

Part 3: Calculate the area when \( x = 3 \)

For Rectangle A:

Substitute \( x = 3 \) into \( 6x^2 + 17x - 3 \):
\( 6(3)^2 + 17(3) - 3 = 6(9) + 51 - 3 = 54 + 51 - 3 = 102 \)

For Rectangle B:

Substitute \( x = 3 \) into \( 21x^2 - 13x + 2 \):
\( 21(3)^2 - 13(3) + 2 = 21(9) - 39 + 2 = 189 - 39 + 2 = 152 \)

Part 4: Compare the areas

Rectangle A’s area: \( 102 \) square units
Rectangle B’s area: \( 152 \) square units

Since \( 152 > 102 \), Rectangle B has a larger area.

Final Answers

• Factored form of Rectangle A: \( \boldsymbol{(6x - 1)(x + 3)} \)
• Factored form of Rectangle B: \( \boldsymbol{(3x - 1)(7x - 2)} \)
• Area of A when \( x = 3 \): \( \boldsymbol{102} \)
• Area of B when \( x = 3 \): \( \boldsymbol{152} \)
• Larger area: Rectangle \( \boldsymbol{B} \)