# find an equation of the parabola with integral coefficients containing the points (1,4), (2,12), and (4,46)

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### 1 Answer

`f(x)= ax^2 + bx + c`

`f(1)= 4`

`==> f(1)= a + b+ c = 4 ..............(1)`

`f(2)= 12 `

`==> f(2)= 4a +2b +c = 12 .................(2)`

`f(4) = 46`

`==> f(4)= 16a +4b +c = 46.............(3)`

Now we have 3 equations with 3variables. Then, we will solve the system using substitution and elimination.

First we will subtract (1) from (2).

==> 3a + b = 8 ..................(4)

Now we will subtract (2) from (3).

==> 12a +2b = 34 ..................(5)

Now we will solve for equations (4) and (5).

==> -2*(4) + (5)

==> 6a = 18

==> a = 18/6 = 3

==> **a = 3**

Now we have a= 3 , so we will substitute into (4) to find b.

==> 3a + b= 8 ==> b= 8-3a = 8-3*3 = 8-9 = -1

==> **b= -1**

Now we will substitute into equation (1) to find c.

==> a + b+ c = 4

==> 3 -1 + c = 4

==> 2+c = 4 ==> **c = 2**

**Then, the equation of the parabola is given by:**

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

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