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.
[tex]QS=RT \ QS=x^2+6x[/tex]
[tex]RT=8x+3[/tex]
[tex]x^2+6x=8x+3[/tex]
[tex]x^2+6x-8x-3=0[/tex]
[tex]x^2-2x-3=0[/tex]
We have obtained a quadratic equation here and solving this gives the possible values of QS.This can be solved using quadratic formula
[tex]x=(-b\pm \sqrt{(b^2-4ac))}/2a[/tex]
The quadratic equation is of the form [tex]ax^2+bxl+c=0[/tex]
here a=1 b=-2 c=-3
[tex]x=(-b\pm \sqrt{(b^2-4ac))}/2a[/tex]
[tex]x=x=(2 \pm \sqrt{ (-2^2-4 \times 1 \times -3))}/(2 \times 1)[/tex]
[tex]x=(2 \pm \sqrt{(4+12))} /2[/tex] or
[tex]x=(2 \pm \sqrt{16)}/2 =(2\pm 4)/2 \ \ \ \ x=(2+4)/2=3 \ \ \ \ or \ \ \ x=(2-4)/2=-1[/tex]
putting x=3 [tex]QS=x^2+6x[/tex]
[tex]=3^2+6 \times 3=9+18=27[/tex]
putting x=-1
[tex]QS=(-1)^2+6 \times -1=1-6=-5[/tex]
Since a length can have only positive values QS=27 units.
