Question

Give possible expressions for the length and breadth of the rectangle whose area is given by,

(i) $25{{a}^{2}}-35a+12$
(ii) \[24{{x}^{2}}-15x\]

Answer 1

Hint: Area of a rectangle is given by,

$Area=length\times breadth$

So, we have to find two expressions, which on multiplication produces the given expression of area. We need to find the factors of the given expression for the area.

Complete step-by-step answer:

In question, expression for area is given. We need to find the expressions which are possible for length and breadth.

So, we need to find two polynomials which when multiplied together produce the expression given for area.

It means we need to split the given area for expression into simpler polynomials.

i.e. we need to find the factors of the given expression for area.

Thus, we will factorize the given polynomial expression for area and get the expressions for length and breadth.

(i) Given expression for area $=25{{a}^{2}}-35a+12$.

As, $length\times breadth=area$ and area is quadratic.

So, the order of polynomials for length and breadth will be less than or equal to 2.

Factorization of $25{{a}^{2}}-35a+12$.

For factorising the above quadratic expression, we need to split the middle term.

$\begin{align}

  & 25{{a}^{2}}-35a+12=25{{a}^{2}}-20a-15a+12 \\

 & =5a\left( 5a-4 \right)-3\left( 5a-4 \right) \\

 & =\left( 5a-3 \right)\left( 5a-4 \right) \\

 & \Rightarrow 25{{a}^{2}}-35a+12=\left( 5a-3 \right)\left( 5a-4 \right) \\

\end{align}$

Thus, for area $=25{{a}^{2}}-35a+12$, we can have following pairs of forms of length and breadth,

Length	Breadth
1	$25{{a}^{2}}-35a+12$
$25{{a}^{2}}-35a+12$	1
$5a-3$	$5a-4$
$5a-4$	$5a-3$

Similarly, we can do for (ii) \[24{{x}^{2}}-15x\].

Let us first factorise \[24{{x}^{2}}-15x\].

Taking $3x$ common, we will get,

\[24{{x}^{2}}-15x=3x\left( 8x-5 \right)\]

Thus, for area \[=24{{x}^{2}}-15x\], we can have the following possible forms of length and breadth,

Length	Breadth
1	\[24{{x}^{2}}-15x\]
\[24{{x}^{2}}-15x\]	1
$3x$	\[8x-5\]
\[8x-5\]	$3x$

Note: A student can make mistakes by not considering the form when length = 1 and breadth is quadratic or length is quadratic and breadth is 1. But these forms will also be taken in answer as taking these forms will also lead to the required area.