SOLUTION: I need help When the range of x determined by {{{-ax^2+bx+4>=0}}} is {{{-1/3<=x<=4}}} then a = ?, b = ?

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: I need help When the range of x determined by {{{-ax^2+bx+4>=0}}} is {{{-1/3<=x<=4}}} then a = ?, b = ?      Log On


   

Question 1040224: I need help
When the range of x determined by
-ax%5E2%2Bbx%2B4%3E=0 is -1%2F3%3C=x%3C=4
then a = ?, b = ?
Found 2 solutions by rothauserc, Edwin McCravy:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
(-a/9) - (b/3) + 4 = 0
-a -3b + 36 = 0
a ÷ 3b - 36 = 0
:
-16a + 4b + 4 = 0
:
We have two equations in 2 unknowns
:
a +3b - 36 = 0
16a - 4b - 4 = 0
:
Solve first equation for a
:
a = -3b + 36
:
Substitute for a in second equation
:
16(-3b+36) -4b -4 = 0
:
-48b +576 -4b -4 = 0
-52b = -572
b = 11
a +33 -36 = 0
a = 3
:
*******
a = 3
b = 11
*******
:

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
When the range of x determined by
-ax%5E2%2Bbx%2B4%3E=0 is -1%2F3%3C=x%3C=4
then a = ?, b = ?
In order for the function 

y+=+-ax%5E2%2Bbx%2B4

to be greater than or equal to 0, its graph must be on or above the
x-axis.  A quadratic equation is the equation of a parabola.  

So this is a parabola that opens downward
and has x-intercepts (real zeros) at -1/3 and 4.





So we are to find the equation of the function

y+=+-ax%5E2%2Bbx%2B4

In order for it to have zeros -1/3 and 4,
it must be of the form

y+=+-a%28x%2B1%2F3%29%28x-4%29 

y+=+-a%28x%5E2-4x%2Bexpr%281%2F3%29x-4%2F3%29

y+=+-ax%5E2%2B4ax-expr%281%2F3%29ax%2Bexpr%284%2F3%29a

y+=+-ax%5E2%2B%284a-expr%281%2F3%29a%29x%2Bexpr%284%2F3%29a

We compare it to

y=-ax%5E2%2Bbx%2B4

The last (constant) terms must be the same, so

expr%284%2F3%29a=4
4a=12
a=3

Substituting in

y+=+-ax%5E2%2B%284a-expr%281%2F3%29a%29x%2Bexpr%284%2F3%29a

y+=+-3x%5E2%2B%284%2A3-expr%281%2F3%29%2A3%29x%2B3%2Aexpr%284%2F3%293

y+=+-3x%5E2%2B%2812-1%29x%2B4

y+=+-3x%5E2%2B11x%2B4

So a = 3 and b = 11

Edwin