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

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step-by-step explanation: