Find the value(s) of d such that 5x^2+5(d-3)x-9d^2+15d+30=0 has two equal roots.

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A quadratic equation cannot have one root. It has to have two roots though the two may be equal. (I have made the change in the question)

Here I guess you want the value of d so that there are two equal roots.

Now for an equation ax^2 + bx + c = 0 to have 2 equal roots b^2 = 4ac.

Now we have

5X^2+5(d-3)X-9d^2+15d+30=0

a= 5 , b = 5(d-3) and c = -9d^2 + 15d + 30

b^2 = 4ac

=> [5(d-3)]^2 = 4(5)(-9d^2+15d + 30)

=> 25(d-3)^2 = 20( -9d^2 +15d + 30)

=> 25(d-3)^2 = 20*3(-3d^2 + 5d + 10)

=> 25(d-3)^2 = 60( -3d^2 + 5d +10)

=> 25(d^2 - 6d + 9) = -180d^2 + 300d + 600

=> 25d^2 - 150d + 225 = -180d^2 + 300d + 600

=> 205d^2 - 450d - 375 = 0

divide by 5

=> 41d^2 - 90d -75 = 0

d1 = [-b + sqrt(b^2 - 4ac)]/2a

=> [45 + 10 sqrt 51] / 41

d2 = [-b - sqrt(b^2 - 4ac)]/2a

=> [45 - 10 sqrt 51] / 41

**The values for d are [45 + 10 sqrt 51] / 41 and [45 - 10 sqrt 51] / 41**

Let f(x) = 5X^2+5(d-3)X-9d^2+15d=30=0

if f(x) has one real root, then Delta must be 0:

==> delta = (b^2 - 4ac)

a = 5 b=5(d-3) c = (-9d^2+15d +30)

==> delta = ( 5(d-3)]^2 - 4(5)( -9d^2+15d + 30) = 0

==> 25(d-3)^2 + 20( 9d2 -15d - 30) = 0

==> 25(d-3)^2 + 20*3(3d^2 - 5d - 10) = 0

==> 25(d-3)^2 + 60( 3d^2 - 5d -10) =

==> 25(d^2 - 6d + 9) + 180d^2 - 300d - 600 = 0

==> 25d^2 - 150d + 225 + 180d^2 - 300d + 600 = 0

==> 205d^2 - 450d - 375 = 0

==> 5( 81d^2 - 90d -75) = 0

==> 5*3( 27d^2 - 30d -25) = 0

==> 15(9d +5) (3d -5) = 0

**==> d1 = -5/9 **

**==> d2= 5/3**

A quadratic equation can always have two roots unless the given roots are equal .

So in case two roots are equal, then the quadratic equation ax^2+bx+c =0 then the discriminant b^2-4ac = 0.

So the given equation 5X^2+5(d-3)X-9d^2+15d+30=0 has the discriminant,

(5(d-3))^2 - 4*5 (-9d^2+15d+30) = 0

25d^2 - 150d +225 +180 d^2 -300d -600 = 0

205 d^2 -30d^2 - 375 = 0.

41d^2 -6d -75 = 0.

d = {6+ sqrt(36+12300)]/2*41 = (6+sqrt12336)/82.

Or d = (6-sqrt12336)/82.

So if d = (6+sqrt12336)/82 or d = (6+sqrt12336)/82, then the given equation has equal roots.

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