b What must the expression's value be given the minus sign?
C
c What must the expression's value be given the minus sign?
D
d The square of a number is always non-negative.
A
a Minimum, x= -4
B
b Maximum, x=-27
C
c Maximum, x=40
D
d Minimum, x= -32
Practice makes perfect
a The square of a number is always non-negative. Therefore, ( x+4)^2 has a minimum value of 0. By setting the base equal to 0, we can determine when this happens by solving for x.
x+4=0 ⇔ x=-4
When x=-4 the expression is minimized.
b As explained in Part A, the square of a number is always non-negative.
(x+27)^2≥ 0
However, since there is a minus sign in front of (x+27)^2, the final result will ultimately be negative.
- (x+27)^2<0
Therefore, - ( x+27)^2 will have a maximum value of 0. By setting the base equal to 0 and solving for x, we can determine when this happens.
x+27=0 ⇔ x=-27
When x=-27 the expression is maximized.
c As in Part B, we have a square that is preceded by a minus sign. Therefore, - ( x-40)^2 will have a maximum value of 0. By setting the base equal to 0, we can determine when this happens by solving for x.
x-40=0 ⇔ x=40
When x=40 the expression is maximized.
d As in Part A, we have a square with no minus sign in front of it. Therefore, ( x+32)^2 can have a minimum value of 0. By setting the base equal to 0, we can determine when this happens by solving for x.
x+32=0 ⇔ x=-32
When x=- 32 the expression is minimized.
