A) Show that 4x² -6x+9 is positive for all values real numbers x.

(4x² -6x+1.5²) +9-1.5²= (2x-1.5)²+9-2.25= (2x-1.5)²+6.75

(2x-1.5)² is greater or equal to 0 , 6.75>0 so expression is positive for all real numbers x

Step-by-step explanation:

a) Use the formula a²-2ab+b²= (a-b)² (You need to make squared expression because it is always positive)

Consider 4x² as a² and -6x=-2ab

4x²=a² a=2x  ab=3x 2x*b=3x

b=1.5

Express 4x² -6x+9= (4x² -6x+1.5²) +9-1.5²= (2x-1.5)²+9-2.25= (2x-1.5)²+6.75

(2x-1.5)² is greater or equal to 0 , 6.75>0 so expression is positive for all real numbers x

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.