Question

*p² / 4 + p / 3 + 1/9 is an expansion of which binomial?*

1️⃣ (p/2 + 1/3)²
2️⃣ (p/2 - 1/3)²
3️⃣ (p/4 + 1/9)²
4️⃣ none of the above​

Answer 1

$\textbf{Given:}$

$\mathsf{\dfrac{p^2}{4}+\dfrac{p}{3}+\dfrac{1}{9}}$

$\textbf{To simplify:}$

$\mathsf{\dfrac{p^2}{4}+\dfrac{p}{3}+\dfrac{1}{9}}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{\dfrac{p^2}{4}+\dfrac{p}{3}+\dfrac{1}{9}}$

$\textsf{This can be written as}$

$\mathsf{=\left(\dfrac{p}{2}\right)^2+\dfrac{p}{3}+\left(\dfrac{1}{3}\right)^2}$

$\mathsf{=\left(\dfrac{p}{2}\right)^2+2{\times}\dfrac{p}{2}{\times}\dfrac{1}{3}+\left(\dfrac{1}{3}\right)^2}$

$\mathsf{Using\;the\;identity,}$

$\boxed{\mathsf{(a+b)^2=a^2+2ab+b^2}}$

$\mathsf{=\left(\dfrac{p}{2}+\dfrac{1}{3}\right)^2}$

$\implies\boxed{\mathsf{\dfrac{p^2}{4}+\dfrac{p}{3}+\dfrac{1}{9}=\left(\dfrac{p}{2}+\dfrac{1}{3}\right)^2}}$

$\textbf{Answer:}$

$\mathsf{Option\;(1)\;is\;correct}$

Answer 2

1️⃣ (p/2 + 1/3)² is the correct answer.

The given expansion is of the basic formula, which is : (a + b)². The formula on solving will be written as : a² + b² + 2*a*b.

Removing the wrong answers first. Second option has negative sign, which isn't in the question. Thus, it can't be the expansion.

Third option and mentioned question has same numbers, which should be squared values for correct answer. Thus, it is wrong option.

Taking first option, the numbers are square values and there is plus sign. Solving the first option then, (p/2)² + (1/3)² + 2*(p/2)*(1/3) = p²/4 + 1/9 + p/3.

Hence, first option (p/2 + 1/3)² is the correct answer.