Determine the intervals for which $4 x ( x - 5 )$ is entirely negative and entirely positive.
A)negative: $( 0,5 )$ positive: $( - \infty , 0 ) , ( 5 , \infty )$
B)negative: $( - \infty , - 5 ) , ( 5 , \infty )$ positive: $( - 5,5 )$
C)negative: $( - \infty , 0 ) , ( 1.25 , \infty )$ positive: $( 0,1.25 )$
D)negative: $( 0,1.25 )$ positive: $( - \infty , 0 ) , ( 1.25 , \infty )$
E)negative: $( - \infty , 0 ) , ( 5 , \infty )$ positive: $( 0,5 )$

Question

Quizplus · Accepted Answer

To determine the intervals for which a quadratic expression is entirely positive or negative, we need to find the roots of the quadratic equation and test the sign of the expression in the intervals defined by the roots. In this case, the quadratic expression is $4x(x-5)$, and it is already factored. The roots are $x=0$ and $x=5$. We can now test the sign of the expression in the intervals defined by the roots. We start with a test point in the interval $(-\infty,0)$: $$4x(x-5)<0$$ $$4(-1)(-1-5)<0$$ $$4\cdot6>0$$ Since the expression is positive for $x=-1$, this means that the interval $(-\infty,0)$ is not entirely negative. We can now test a point in the interval $(0,5)$: $$4x(x-5)<0$$ $$4(1)(1-5)<0$$ $$4\cdot(-4)<0$$ Since the expression is negative for $x=1$, this means that the interval $(0,5)$ is entirely negative. Finally, we test a point in the interval $(5,\infty)$: $$4x(x-5)<0$$ $$4(6)(6-5)<0$$ $$4>0$$ Since the expression is positive for $x=6$, this means that the interval $(5,\infty)$ is not entirely negative. Therefore, the intervals for which the expression $4x(x-5)$ is entirely negative and entirely positive are: $$ ext{negative: } (0,5)$$ $$ ext{positive: } (-\infty,0), (5,\infty)$$ The only choice that matches this result is A.

Determine the Intervals for Which 4x(x−5)4 x ( x - 5 )4x(x−5)

Determine the Intervals for Which $4 x ( x - 5 )$