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The parabola $y = ax^2 + bx + c$ is graphed below. Find $a \cdot b \cdot c.$ (The grid lines are one unit apart.)
The graph passes through (-3,5), (-1,7), and (2,-1).

 Mar 9, 2024
 #1
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a(-3)^2  - (3)b + c =  5

a(-1)^2  - 1b + c = 7

a(2)^2 + 2b + c  = -1

 

9a  - 3b + c =  5     (1)

1a  -1b + c   = 7  →  -1a + 1b - c  = -7     (2)

4a  + 2b + c  = -1    (3)

 

Add (1),(2)

8a -2b = -2    →  4a - b = -1 →  b = 4a + 1    (3)

 

Add (2) , (3)

3a + 3b = -8       (4)

 

Sub (3)  into 4

3a + 3(4a + 1) = -8

15a + 3   = - 8

15a  = -11

a = -11/15

 

b = 4(-11/15)+ 1 =  -29/15

 

a - b + c   =  7

(-11/15) - (-29/15) + c  =7

18/15 + c  = 7

c = 105/15 - 18/15 = 87/15   = 29/5

 

a* b * c   =   (-11/15)(-29/15)(29/5)   =    9251  /  1125

 

cool cool cool

 Mar 9, 2024

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