# Let f(x) = mx^2 + 2x + 5 and g(x) = 2x^2 - nx - 2. The functions are combined to form the new function h(x) = f(x) X g(x). Points (1,-40) &... (-1,24) satisfy the new function. Determine f(X) and g(x).

## Expert Answers

`f(x)= mx^2 + 2x + 5`

`g(x)= 2x^2 - nx -2`

`h(x)= f(x) X g(x)= (mx^2 + 2x +5)(2x^2-nx -2) `

`h(1)= (m+2+5)(2-n-2) = -40 ==> (m+7)(-n)= -40`

`==>(m+7)(n) = 40............(1)`

`h(-1)= (m-2+5)(2+n -2) = 24==> (m+3)(n)= 24`

`==> (m+3)(n) = 24 .............(2)`

We will divide (1) by (2):

`==>...

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`f(x)= mx^2 + 2x + 5`

`g(x)= 2x^2 - nx -2`

`h(x)= f(x) X g(x)= (mx^2 + 2x +5)(2x^2-nx -2) `

`h(1)= (m+2+5)(2-n-2) = -40 ==> (m+7)(-n)= -40`

`==>(m+7)(n) = 40............(1)`

`h(-1)= (m-2+5)(2+n -2) = 24==> (m+3)(n)= 24`

`==> (m+3)(n) = 24 .............(2)`

We will divide (1) by (2):

`==> (m+7)/(m+3)= 40/24`

`==> (m+7)/(m+3)= 5/3`

`==> 3(m+7)= 5(m+3)`

`==> 3m +21 = 5m+15`

`==> 2m = 6`

`==> m= 3`

`==> (m+3)n = 24`

`==> 6n = 24 ==> n = 24/6 = 4`

`==> f(x)=3x^2+2x + 5`

`==>g(x)= 2x^2 -4x -2`

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