# Express equation (18x^2-12x)/(2x^2-4x+2)=a in the general form of a quadratic equation, then determine the value of a such that the equation has two distinct real roots

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Multiply both sides of the equation by the denominator on the lefthand side giving

`18x^2 - 12x = a(2x^2 - 4x + 2)`

Move all terms to the righthand side of the equation giving

`a(2x^2 - 4x + 2) - 18x^2 - 12x = 0`

Multiply out the bracket

`2ax^2 - 4ax + 2a - 18x^2 - 12x = 0`

Gather terms

`(2a-18)x^2 -(4a+12)x + 2a = 0`

` `Divide both sides by 2

`(a-9)x^2 - (2a + 6)x + a = 0`

`x = (-B +- sqrt(B^2 - 4AC))/(2A)`

where `A= (a-9)`, `B = -(2a + 6)` and `C = a`

Gives

`x = ((2a+6) +-sqrt((2a+6)^2 - 4(a-9)a))/(2(a-9))`

For the quadratic to have 2 distinct real roots the determinant must be greater than zero

ie, `(2a+6)^2 - 4a(a-9) > 0`

Multiplying out brackets

`4a^2 + 24a + 36 - 4a^2 +36a > 0`

Gathering terms

`60a + 36 > 0`

`implies ` `60a > -36` `implies a> -3/5`

a > -3/5

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