Can You Help? MC=150-40Q+3Q^2 And AVC=150-20Q+Q^2 .

MC=150-40Q+3Q^2
AVC=150-20Q+Q^2 .

Average variable cost will be minimized at the quantity where average variable cost equals marginal cost.

MC = AVC

150-40Q+3Q^2 = 150-20Q+Q^2
150-40Q+3Q^2-150+20Q-Q^2 =0

Grouping the like terms
(150- 150)-(40Q-20Q) +(3Q^2-Q^2) =0
20 Q + 2Q^2  = 0
2Q^2  – 20 Q = 0

Taking 2 as common
2(Q^2  – 10 Q )=0

Dividing both sides by 2you will get
Q^2  – 10 Q = 0
Q^2  = 10 Q

Dividing both sides by Q you will get
Q = 10

Put the value of Q in 150-20Q+Q^2 you will get
150-20Q+Q^2
= 150-20(10)+(10)^2
=150-200+100
=150 – 100
= 50

So minimum value of P is 50