Answer:
(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
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