A rectangular pool has area 2x2 + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length. Hasta was a unit used to measure length.
Step-by-step solution
To find: Length of the pool
Idea: Area of a rectangle = length × width. So factorise the area; one factor is the width 2x + 1 and the other factor is the length. (A hasta is an old Indian unit of length, roughly the distance from the elbow to the fingertip.)
- Area = length × width, so we need 2x2 + 7x + 3 = (2x + 1) × length. Factorise the area.½ mark
- Split the middle term 7x. Product of first and last coefficients = 2 × 3 = 6. Two numbers with product 6 and sum 7 are 6 and 1.
2x2 + 7x + 3 = 2x2 + 6x + x + 31 mark - = 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)1 mark - One factor, 2x + 1, is the width. So the length = (x + 3) hastas.½ mark
Check: Multiply back: (2x + 1)(x + 3) = 2x2 + 6x + x + 3 = 2x2 + 7x + 3 ✓. With x = 2: area = 8 + 14 + 3 = 25, width = 5, length = 5, and 5 × 5 = 25 ✓.
Answer to write in the exam
Area = 2x2 + 7x + 3 = 2x2 + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)
Length = AreaWidth = (2x + 1)(x + 3)2x + 1
∴ Length of the pool = (x + 3) hastas
Common mistakes that cost marks
- Choosing 3 and 4 (sum 7) to split 7x. The two numbers must also multiply to 2 × 3 = 6, so they are 6 and 1.
- Giving (2x + 1) as the answer. That is the width; the length is the other factor.
- Writing the unit as square hastas. Length is measured in hastas; only the area is in square hastas.
How this can come in the exam
A rectangle has area 3x2 + 11x + 6 square units and width 3x + 2 units. Its length is
- x + 2
- x + 3
- 3x + 3
- x + 6
Show answer
(B) x + 3
3x2 + 11x + 6 = 3x2 + 9x + 2x + 6 = (3x + 2)(x + 3), so the length is x + 3.
Assertion (A): If a rectangle has area 2x2 + 7x + 3 and width 2x + 1, its length is x + 3.
Reason (R): 2x2 + 7x + 3 = (2x + 3)(x + 1).
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(C) A is true but R is false.
A is true, since 2x2 + 7x + 3 = (2x + 1)(x + 3). R is false: (2x + 3)(x + 1) = 2x2 + 5x + 3.
Try one yourself
A rectangular garden has area 6x2 + 7x + 2 square metres and width (2x + 1) m. Find its length.
Show answer
6x2 + 4x + 3x + 2 = 2x(3x + 2) + 1(3x + 2) = (2x + 1)(3x + 2). Length = (3x + 2) m.
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