SOLUTION: A quadratic equation has a parable that has the roots x = 3 and x = -5. It passes through the point (1, -12). Determine its equation
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Question 764034: A quadratic equation has a parable that has the roots x = 3 and x = -5. It passes through the point (1, -12). Determine its equation
Found 2 solutions by solver91311, ankor@dixie-net.com:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Really? I have heretofore never encountered a quadratic equation that had a normative principle to teach. Is it anything like the parable of the Prodigal Son or perhaps more like the Good Samaritan?
Now, if what you meant to say was that the graph of a quadratic function is a parabola and the function has zeros at 
and 
and that the graph contains the point )
. Then it is possible to derive the equation that defines the function.
The standard form of a quadratic function is \ =\ ax\^2\ +\ bx\ +\ c)
If the function has a zero at , then the point )
must be on the graph. Therefore:
\^2\ +\ b(3)\ +\ c\ =\ 0)
Likewise, for
\^2\ +\ b(-5)\ +\ c\ =\ 0)
and in the case of the point :
\^2\ +\ b(1)\ +\ c\ =\ -12)
Leading us to the following 3X3 system
which translates to the following augmented matrix, the ordered triple solution of which provides the coefficients for your desired function.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A quadratic equation has a parabola that has the roots x = 3 and x = -5.
It passes through the point (1, -12).
Determine it's equation.
:
Using the form ax^2 + bx + c = y, write equation for each pair
x=3; y = 0: 9a + 3b + c = 0
x=-.5; y=0: .25a - .5b + c = 0
x=1; y=-12: a + b + c = -12
:
Subtract the 3rd equation from the 1st equation
9a + 3b + c = 0
a + b + c = -12
-----------------
8a + 2b = 12
simplify, divide by 2
4a + b = 6
:
Subtract the 3rd equation from the 2nd equation
.25a - .5b + c = 0
a + b + c = -12
----------------
-.75a - 1.5b = 12
Simplify, divide both sides by -.75
a + 2b = -16
:
Two 2 unknown equations we can solve with elimination
Multiply the 1st by 2, subtract the 2nd
8a + 2b = 12
a + 2b = -16
-----------------subtraction eliminates b, find a
7a = 28
a = 28/7
a = 4
:
Find b using a + 2b = -16
4 + 2b = -16
2b = -20
b = -10
:
Find c
4 - 10 + c = -12
c = -12 + 6
c = -6
:
Our equation: y = 4x^2 - 10x - 6
:
Graphically
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