Question

In rectangle QRST, QS = x2 + 6x, and RT = 8x + 3. Find the length of QS.

Answer 1

The length of QS is 27 units.

Step-by-step explanation:

In the given rectangle QRST QS and RT are the diagonals. In a triangle, the opposite sides are equal and parallel. This implies that the diagonals of a rectangle are equal.

$QS=RT \ QS=x^2+6x$

$RT=8x+3$

$x^2+6x=8x+3$

$x^2+6x-8x-3=0$

$x^2-2x-3=0$

We have obtained a quadratic equation here and solving this gives the possible values of QS.This can be solved using quadratic formula

$x=(-b\pm \sqrt{(b^2-4ac))}/2a$

The quadratic equation is of the form $ax^2+bxl+c=0$  

here a=1 b=-2 c=-3  

$x=(-b\pm \sqrt{(b^2-4ac))}/2a$

$x=x=(2 \pm \sqrt{ (-2^2-4 \times 1 \times -3))}/(2 \times 1)$

$x=(2 \pm \sqrt{(4+12))} /2$ or  

$x=(2 \pm \sqrt{16)}/2 =(2\pm 4)/2 \ \ \ \ x=(2+4)/2=3 \ \ \ \ or \ \ \ x=(2-4)/2=-1$

putting x=3 $QS=x^2+6x$

$=3^2+6 \times 3=9+18=27$

putting x=-1

$QS=(-1)^2+6 \times -1=1-6=-5$

Since a length can have only positive values QS=27 units.