SOLUTION: If the zeroes of a quadratic function are -4 and 6, and its y-intercept is 12, determine the function in general form without using regression. My name is Chantal and I know that

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Question 187247: If the zeroes of a quadratic function are -4 and 6, and its y-intercept is 12, determine the function in general form without using regression.
My name is Chantal and I know that -4 and 6 are the roots and -12 is the last part of the equation in general form but I don't know what he means by regression. I'm not sure how to go about doing this equation. If you could help me it would be greatly appreciated. I missed school when we learnt this.This is not from a book.
Chantal Laroche
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"If the zeroes of a quadratic function are -4 and 6" means that the solutions to the equation ax%5E2%2Bbx%2Bc=0 are x=-4 or x=6 (ie if you plug in x=-4 or x=6 into ax%5E2%2Bbx%2Bc, you will get zero)




x=-4 or x=6 Start with the given solutions


x%2B4=0 or x-6=0 Get the terms to the left side (so the right side is zero)


%28x%2B4%29%28x-6%29=0 Combine the equations using the zero product property


Note: the zero product property states that if A%2AB=0 then either A=0 or B=0 (or possibly both)


a%28x%2B4%29%28x-6%29=0 Place an additional variable "a" onto the front of the equation. Note: this will be needed to satisfy the third condition.


a%28x%5E2-6x%2B4x-24%29=0 FOIL


a%28x%5E2-2x-24%29=0 Combine like terms.


So the equation that has the zeros of -4 and 6 is y=a%28x%5E2-2x-24%29


Now there's an additional condition: The y-intercept is 12


This means that when x=0, y=12


y=a%28x%5E2-2x-24%29 Start with the previous equation.


12=a%280%5E2-2%280%29-24%29 Plug in x=0 and y=12


12=a%280-2%280%29-24%29 Square 0 to get 0


12=a%280-0-24%29 Multiply


12=a%28-24%29 Combine like terms.


12%2F%28-24%29=a Divide both sides by -24 to isolate "a".


-1%2F2=a Reduce


So the value of "a" is a=-1%2F2


This means that the equation then goes from y=a%28x%5E2-2x-24%29 to y=%28-1%2F2%29%28x%5E2-2x-24%29


Now distribute to get: y=%28-1%2F2%29%28x%5E2%29-%28-1%2F2%292x-%28-1%2F2%2924


and multiply: y=%28-1%2F2%29x%5E2%2Bx%2B12


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Answer:


So the equation with the zeros -4 and 6 with the y-intercept of 12 is:


y=%28-1%2F2%29x%5E2%2Bx%2B12


Here's the graph to visually verify our answer:




Graph of y=%28-1%2F2%29x%5E2%2Bx%2B12 with x-intercepts (-4,0) and (6,0) with the y-intercept (0,12)