[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
- Solve the following word problems.
[tex] \large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}[/tex]
Question 1 ↴
➜ Area of the lot = 2x² + 7x + 3 cm²
➜ Width of the garden = 2x + 1 cm.
➜ Length of the garden = y
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✪ Area of a rectangle = length × width
⇒ length = area ÷ width
⇒ y = 2x² + 7x + 3 ÷ 2x + 1
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WORKING ↷
[tex] \tt \: y = \frac{ {2x}^{2} + 7x + 3}{2x + 1} \\ \\ \sf \: Factorise \: {2x}^{2} + 7x + 3. \\ \\ \tt \: y = \frac{\left(x+3\right)\left(2x+1\right)}{2x+1} \\ \\ \sf \: Cancel \: out \:( 2x + 1 )\\ \\ \large\boxed{\boxed{\bf y = x+3 }}[/tex]
✯ Length of the garden = x + 3 cm.
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Question 2 ↴
➜ Area of the frame = 4x² - 4xy + y² cm²
➜ Length of the side of the frame = s
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✪ Area of a square = side²
⇒ 4x² - 4xy + y² = s²
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WORKING ↷
[tex] \tt {4x}^{2} - 4xy + {y}^{2} = {s}^{2} \\ \\ \sf \: Use \: the \: algebraic \: identity \downarrow \: \\ \sf {a}^{2} - 2ab + {b}^{2} = (a - b) ^{2} ... \\ \sf \: a = 2x \: and \: b = y \\ \\ \tt \left(2x-y\right)^{2} = {s}^{2} \\ \\ \sf \: Squaring \: on \: both \: the \: sides \\ \\ \tt \sqrt{(2x - y) ^{2} } = \sqrt{ {(s)}^{2} } \\ \large\boxed{\boxed{\bf \: (2x - y) = s}}[/tex]
✯ Length of the side of the frame = 2x - y cm.
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Question 3 ↴
➜ Length of the rectangle = x + 5 cm
➜ Width of the garden = x + 3 cm.
➜ Area of the garden = a
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✪ Area of a rectangle = length × width
⇒ a = (x + 5) × (x + 3)
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WORKING ↷
[tex] \tt \: a = (x + 5) \times (x + 3) \\ \\ \sf \: multiply \: (x + 5) \: with \: (x + 3) \\ \\ \tt \: a = (x + 5) \times (x + 3) \\ \tt \: a = x(x + 3) + 5(x + 3) \\ \tt \: a = {x}^{2} + 3x + 5x + 3 \\ \large \boxed{\boxed{ \bf \: a = {x}^{2} + 8x + 3}}[/tex]
✯ Area of the rectangle = x² + 8x + 3 cm².
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Question 4 ↴
➜ Length of the side of a square = x + 6 cm
➜ Area of the square = a
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✪ Area of a square = side × side
⇒ Area of a square = side²
⇒ a = (x + 6)²
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WORKING ↷
[tex] \tt \: a = (x + 6) ^{2} \\ \\ \sf \: Use \: the \: algebraic \: identity \downarrow \: \\ \sf (a + b) ^{2} = {a}^{2} + 2ab + {b}^{2} ... \\ \sf \: a = x \: and \: b = 6 \\ \\ \tt \: a = (x + 6) ^{2} \\ \tt \: a = {x}^{2} + 2 \times x \times 6 + {6}^{2} \\ \large \boxed{\boxed{ \bf \: a = {x}^{2} + 12x + 36}}[/tex]
✯ Area of the square = x² + 12x + 36 cm².
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