Question

Jane bought a new rectangular placemat with an area of 126 square inches. The length of the placemat is four times the quantity of nine less than half its width. Complete the equation that models the area of the placemat, in terms of the width of the placemat, w.

Answer 1

let
x------> the length of the placemat
w------> the width of the placemat

we know that
area of the rectangular placemat=x*w
Area=126 in²
so
126=x*w-----> x=126/w ----> equation 1
x=4*[9-w/2]----> x=4*[(18-w)/2]----> x=36-2w----> equation 2
equate equation 1 and equation 2
126/w=36-2w----> multiply by w both sides----> 126=36w-2w²
2w²-36w+126=0

using a graph tool
see the attached figure

the solutions are
w=4.757 in
w=13.243 in

for w=4.76 in
x=126/w----> x=126/4.76----> x=26.47 in

for w=13.24 in
x=126/13.24-----> x=9.52 in

the answer is
the equation that models the area of the placemat is
2w²-36w+126=0