Question

How do you evaluate the function `f (x) = 3x^2 + 3x -2` for `f (2/a)`?

Answer 1

By 'evaluate' I assume you mean find `a` for `f(2/a)=0`
`color(green)(a=-1+- sqrt(7))`

Explanation:

Let `color(white)(xx)y_1=3x^2+3x-2 ->f(x)`

Then `y_1=3(2/a)^2+3(2/a)-2 -> f(2/a)`

`y_1=3(4/a^2) +3(2/a)-2`

`y_1=12/a^2+6/a-2`

Let `color(white)(xx)y_2 =1/y_1` giving

`y_2=1/12a^2+1/6a -1/2 =0`

Multiply throughout by 12 giving:

`y_2=a^2+2a-6=0`
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`color(green)("Using std form"color(white)(xxx) ax^2+bx+c=0)`

`color(green)("Where" color(white)(xx)x = (-b+-sqrt(b^2-4ac))/(2a)`

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In the context of the given question we have:

`a=(-2+-sqrt((2)^2-4(1)(-6)))/(2(1))`

`a=(-2+-sqrt(4+24))/2`

`a=-1+- sqrt(7)`

`a~~ -1+- 2.65` to 2 decimal places

`a~~-3.65color(white)(x) :color(white)(x) a~~1.65`