Question

Calculate the length and breadth of rectanglr whose area is 2xsquare-7x+15

Answer 1

Given:

A rectangle with area of 2x²-7x+15

To Find:

Length and breadth of given rectangle

Concept to be used: Since, the area is in the form of quadratic polynomial, so it will have two factors. Therefore, after factorising the area, the two factors so obtained will be the length and breadth of the given rectangle

Solution:

We know that,

According to quadratic formula, roots of quadratic equation of the form of ax²+bx+c=0 is

$\longrightarrow\bf{x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}}$

Now, the given quadratic polynomial is

➳ 2x²-7x+15

Let x be the root of the given polynomial

Then,

➳ 2x²-7x+15= 0

On applying quadratic formula in the above equation, we get

$\longrightarrow\mathrm{x=\dfrac{-(-7)\pm\sqrt{(-7)^{2}-4(2)(15)}}{2(2)}}$

$\longrightarrow\mathrm{x=\dfrac{7\pm\sqrt{49-120}}{4}}$

$\longrightarrow\mathrm{x=\dfrac{7\pm\sqrt{-71}}{4}}$

$\longrightarrow\mathrm{x=\dfrac{7\pm i\sqrt{71}}{4},where\:i=\sqrt{-1}}$

$\longrightarrow\mathrm{x=\dfrac{7+i\sqrt{71}}{4},\dfrac{7-i\sqrt{71}}{4}}$

$\longrightarrow\mathrm{\bigg(x-\dfrac{7+i\sqrt{71}}{4}\bigg)\bigg(x-\dfrac{7-i\sqrt{71}}{4}\bigg)=0}$

Therefore, given rectangle has no real length and breadth but have imaginary length and breadth.

where,

$\longrightarrow\mathrm{\pink{Length=x-\dfrac{7+i\sqrt{71}}{4}}}$

$\longrightarrow\mathrm{\green{Breadth=x-\dfrac{7-i\sqrt{71}}{4}}}$