Question

Please help !! 30 POINTS
The following equations model the number of bikes sold at two different bike shops, where x is the number of months since the shops opened. in which month(s) did both shops sell the same number of bikes? how many bikes did each shop sell in those months?

Shop A : y = -x^2 + 40x
Shop B : y = 18x + 72

Answer 1

$Shop\ A:y=-x^2+40x\\\\Shop\ B:y=18x+72\\\\-x^2+40x=18x+72\qquad\text{subtract 18x from both sides}\\\\-x^2+22x=72\qquad\text{change the signs}\\\\x^2-22x=-72\\\\x^2-2(x)(11)=-72\qquad\text{add}\ 11^2\ \text{to both sides}\\\\x^2-2(x)(11)+11^2=-72+11^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-11)^2=-72+121\\\\(x-11)^2=49\to x-11=\pm\sqrt{49}\\\\x-11=-7\ \vee\ x-11=7\qquad\text{add 11 to both sides}\\\\x=4\ \vee\ x=18\\\\For\ x=4\to y=18(4)+72=144\\For\ x=18\to y=18(18)+72=396$

$Answer:\\\\Number\ of\ months=4,\ number\ of\ bikes=144\\\\Number\ of\ months=18,\ number\ of\ bikes=396$