Answer:
3/5
Step-by-step explanation:
First one 18^2 - 4(-1)(81) > 0 so no repeated root
Second one (-3)^2 - 4(3) (-168) > 0 so no repeated root
Third one (-4) - (4) (1) (-4) > 0 so no repeated root
Fourth one (-30)^2 - 4(25) (9) = 900 - 100(9) = 0 repeated root
Note this factors as (5x - 3)^2
(5x - 3)^2 = 0 take both roots
5x - 3 = 0
5x = 3
x = 3/5 = repeated root
The quadratic expression [tex]25x^{2} -30x+9[/tex] has repeated root and the repeated root is 3/5
Let, [tex]ax^{2} +bx+c=0[/tex] is a quadratic.
Then the expression ([tex]b^{2}-4ac[/tex]) is called the discriminant of the quadratic.
If [tex]b^{2}-4ac[/tex] = 0 then we can say that the quadratic has a repeated real root.
The discriminant of [tex]-x^{2} +18x+81[/tex] is,
[tex]b^{2}-4ac[/tex] = [tex]18^{2}-4(-1)(81)[/tex] = 324+324=648 > 0
So, This quadratic has no repeated root.
The discriminant of [tex]3x^{2} -3x-168[/tex] is,
[tex]b^{2}-4ac[/tex] = [tex](-3)^{2}-4(3)(-168)[/tex] = 9+2016 = 2025 > 0
So, This quadratic has no repeated root.
The discriminant of [tex]x^{2} -4x-4[/tex] is,
[tex]b^{2}-4ac[/tex] = [tex](-4)^{2}-4(1)(-4)[/tex] = 16+16 = 32 > 0
So, This quadratic has no repeated root.
The discriminant of [tex]25x^{2} -30x+9[/tex] is,
[tex]b^{2}-4ac[/tex] = [tex](-30)^{2} -4(25)(9)[/tex] = 900-900 = 0
So, This quadratic has repeated root.
The discriminant of [tex]x^{2} -14x+24[/tex] is,
[tex]b^{2}-4ac[/tex] = [tex](-14)^{2}-4(1)(24)[/tex] = 196-96 = 100 > 0
So, This quadratic has no repeated root.
The equation [tex]25x^{2} -30x+9[/tex] = 0 has repeated root.
Simplifying the equation we get,
[tex]25x^{2} -30x+9[/tex] = 0
⇒ [tex](5x)^{2}-2(5x)(3)+3^{2}[/tex] = 0
⇒ [tex](5x-3)^{2}[/tex] = 0
⇒ 5x-3 = 0
⇒ 5x = 3
⇒ x = 3/5
The repeated root is 3/5.
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