SOLUTION: Foil method question x^2-106x+480 what two numbers add to get 106, and multiply to get 480?

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Question 781798: Foil method question
x^2-106x+480
what two numbers add to get 106, and multiply to get 480?
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Foil method question
x^2-106x+480
what two numbers add to get 106, and multiply to get 480?
------
Inverse foil is used when the conditions you listed
can be met with integers.
----
Your problem does not lend itself to that prcess.
You have to use the Quadratic formula to find
the zeroes of your trinomia, then create the factor
forms needed to factor the problem.
----
Your zeroes are 4.7403... and x = 101.1297...
Then the factors would be (x-4.703..)(x-101.1297...)
---------------
Cheers,
Stan H.
===============

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

two numbers that you can add to get 106 are 53 and 53 (when you factor 106 you get 2 and 53),
but, there is no number that you can multiply by 53 to get 480; so, you need to complete the square this way:

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=1+x%5E2-106+x%2B480 Start with the given equation



y-480=1+x%5E2-106+x Subtract 480 from both sides



y-480=1%28x%5E2-106x%29 Factor out the leading coefficient 1



Take half of the x coefficient -106 to get -53 (ie %281%2F2%29%28-106%29=-53).


Now square -53 to get 2809 (ie %28-53%29%5E2=%28-53%29%28-53%29=2809)





y-480=1%28x%5E2-106x%2B2809-2809%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 2809 does not change the equation




y-480=1%28%28x-53%29%5E2-2809%29 Now factor x%5E2-106x%2B2809 to get %28x-53%29%5E2



y-480=1%28x-53%29%5E2-1%282809%29 Distribute



y-480=1%28x-53%29%5E2-2809 Multiply



y=1%28x-53%29%5E2-2809%2B480 Now add 480 to both sides to isolate y



y=1%28x-53%29%5E2-2329 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=53, and k=-2329. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=1x%5E2-106x%2B480 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-106x%2B480%29 Graph of y=1x%5E2-106x%2B480. Notice how the vertex is (53,-2329).



Notice if we graph the final equation y=1%28x-53%29%5E2-2329 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-53%29%5E2-2329%29 Graph of y=1%28x-53%29%5E2-2329. Notice how the vertex is also (53,-2329).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.