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`
Solve using the quadratic formula
`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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