Question

the area of a rectangular plot is 528 \text{ m}^2. the length of the plot is one more than twice its width. find the length and the width of the plot.

solution
let \\(l\\) be the length and \\(w\\) be the width of the plot, respectively.
the length of the plot is given as one more than twice its width. i.e.,
\\(l = 1 + 2w\\) -----------------(1)
given: area of the rectangular plot is 528 \text{ m}^2.
\\(lw = 528\\) -----------------(2)
on replacing equation (1) in equation (2),
\\((1 + 2w)w = 528\\)
\\(w + 2w^2 = 528\\)
\\(w + 2w^2 - 528 = 0\\)

value: 2
find the factors of the above quadratic equation.
mark all correct answers
a. \\((w - 16)\\)
b. \\((w + 33)\\)
c. \\((2w + 33)\\)
d. \\((w + 16)\\)
e. \\((w - 33)\\)
f. \\((2w - 33)\\)

Explanation:

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<reasoning>

Set up the quadratic equation

Using the Quadratic Equations knowledge point
\[
2w^2 + w - 528 = 0
\]

Find the factors by splitting the middle term

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &2w^2 + 33w - 32w - 528 = 0\\ &w(2w + 33) - 16(2w + 33) = 0\\ &(w - 16)(2w + 33) = 0 \end{aligned}$$

\]

Identify the correct factor options

Using the Quadratic Equations knowledge point
\[
\text{Factors: } (w - 16) \text{ and } (2w + 33)
\]
</reasoning>

<answer>
<mcq-correct>a. (w - 16)</mcq-correct>
<mcq-option>b. (w + 33)</mcq-option>
<mcq-correct>c. (2w + 33)</mcq-correct>
<mcq-option>d. (w + 16)</mcq-option>
<mcq-option>e. (w - 33)</mcq-option>
<mcq-option>f. (2w - 33)</mcq-option>
</answer>

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Answer:

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<reasoning>

Set up the quadratic equation

Using the Quadratic Equations knowledge point
\[
2w^2 + w - 528 = 0
\]

Find the factors by splitting the middle term

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &2w^2 + 33w - 32w - 528 = 0\\ &w(2w + 33) - 16(2w + 33) = 0\\ &(w - 16)(2w + 33) = 0 \end{aligned}$$

\]

Identify the correct factor options

Using the Quadratic Equations knowledge point
\[
\text{Factors: } (w - 16) \text{ and } (2w + 33)
\]
</reasoning>

<answer>
<mcq-correct>a. (w - 16)</mcq-correct>
<mcq-option>b. (w + 33)</mcq-option>
<mcq-correct>c. (2w + 33)</mcq-correct>
<mcq-option>d. (w + 16)</mcq-option>
<mcq-option>e. (w - 33)</mcq-option>
<mcq-option>f. (2w - 33)</mcq-option>
</answer>

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"Quadratic Equations"
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