Final Answer:
The values of x that would support Daria's claim that the expression –45x2 – x + 2.5 is sometimes negative are A. –5/4 and D. –1.84.
Step-by-step explanation:
To ascertain whether the expression –45x² – x + 2.5 is sometimes negative, we'll assess the given values of x.
Starting with x = –5/4:
Substituting –5/4 into the expression:
–45 * (–5/4)² – (–5/4) + 2.5
First, let's evaluate the values:
–45 * (–5/4)² = –45 * 25/16 = –1125/16
–(–5/4) = 5/4
So, –1125/16 + 5/4 + 2.5 = –1125/16 + 20/16 + 40/16 = –1125/16 + 60/16 = –1065/16
The result, –1065/16, is negative, confirming that –5/4 satisfies the condition that makes the expression negative.
Moving on to x = –1.84:
Substituting –1.84 into the expression:
–45 * (–1.84)² – (–1.84) + 2.5
Evaluating the values:
–45 * (–1.84)² = –45 * 3.3856 = –152.352
–(–1.84) = 1.84
Now adding these up: –152.352 + 1.84 + 2.5 = –152.352 + 4.34 = –148.012
As –148.012 is also negative, –1.84 satisfies the condition for the expression to be negative.
Thus, both –5/4 and –1.84 are values for x that produce negative results in the expression, supporting Daria's claim.
Conversely, when substituting 1 and 2.3 into the expression:
For x = 1: –45 * 1² – 1 + 2.5 = –45 - 1 + 2.5 = –43.5 + 2.5 = –41 which is positive.
For x = 2.3: –45 * 2.3² – 2.3 + 2.5 = –45 * 5.29 – 2.3 + 2.5 = –238.05 – 2.3 + 2.5 = –238.05 – 0.8 = –238.85 which is also positive.
Therefore, options A) –5/4 and D) –1.84 are the values of x that support Daria's claim, while options B) 1 and C) 2.3 do not satisfy the condition of making the expression negative.