A rectangular pool has area square hastas. If its width is hastas, find its length. Hasta was a unit used to measure length.

Understand the Question The area of a rectangle is given by the formula . We are given: To find the length, we divide the area by the width: , which can be solved by factorising the quadratic polynomial . Step 1 · Factorise the Quadratic Expression for Area To factorise , we find two numbers whose product is and whose sum is . The numbers are and . Step 2 · Calculate the Length Using the relationship Answer > Common Mistakes Splitting the Middle Term Error: Looking for factors of instead of the product . Omitting the Unit: Forgetting to write the unit (hastas) in the final answer.

Exploring Algebraic Identities | EOT

Question 9

A rectangular pool has area $2x^2 + 7x + 3$ square hastas. If its width is $2x + 1$ hastas, find its length. Hasta was a unit used to measure length.

Solution

Understand the Question

The area of a rectangle is given by the formula $\text{Area} = \text{Length} \times \text{Width}$ .
We are given:
- $\text{Area} = 2x^2 + 7x + 3$
- $\text{Width} = 2x + 1$
To find the length, we divide the area by the width: $\text{Length} = \dfrac{\text{Area}}{\text{Width}}$ , which can be solved by factorising the quadratic polynomial $2x^2 + 7x + 3$ .

Step 1 · Factorise the Quadratic Expression for Area

To factorise $2x^2 + 7x + 3$ , we find two numbers whose product is $2 \times 3 = 6$ and whose sum is $7$ . The numbers are $6$ and $1$ .

\begin{aligned} 2x^2 + 7x + 3 &= 2x^2 + 6x + x + 3 \\ &= 2x(x + 3) + 1(x + 3) \\ &= (2x + 1)(x + 3) \end{aligned}

Step 2 · Calculate the Length

Using the relationship $\text{Area} = \text{Length} \times \text{Width}$

\begin{aligned} \text{Length} &= \dfrac{\text{Area}}{\text{Width}} \\[0.6em] &= \dfrac{(2x + 1)(x + 3)}{2x + 1} \\[0.6em] &= x + 3 \end{aligned}

Answer

$(x + 3) \text{ hastas}$

Common Mistakes

Splitting the Middle Term Error: Looking for factors of $3$ instead of the product $a \times c = 2 \times 3 = 6$ .
Omitting the Unit: Forgetting to write the unit (hastas) in the final answer.

More questions in EOT

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Q10

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Q11

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Q12

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Q13

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