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Of the five quadratics listed below, four of them have two distinct roots. The fifth quadratic has a repeated root. Find the value of the repeated root.


\( \begin{align*} &-x^2 + 18x + 81 \\ &3x^2 - 3x - 168 \\ &x^2 - 4x - 4 \\ &25x^2 - 30x + 9 \\ &x^2 - 14x + 24 \end{align*}\)

 

:D

 Apr 7, 2020
 #1
avatar+932 
+1

The first quadratic does not factor so we can skip that one for now...

The second quadratic factors into 3(x+7)(x−8) which doesn't have any repeating roots so this is not the answer.

The third quadratic does not factor so we can skip that one for now...

The fourth quadratic factors into (5x−3)(5x−3) which has repeating roots so this is our answer.

The fifth quadratic factors into (x−2)(x−12) which doesn't have any repeating roots so this is not the answer.

 

 

Hope it helps!

 Apr 7, 2020
 #2
avatar+111329 
+1

Thanks, HELPMEEEEEEEEEEEEE    !!!!

 

Here  is  another  method

 

If the discriminant  = 0   we  have a repeated root

 

(1)  (18)^2 - 4(-1)(81)  > 0      NO

 

(2)  (-3)^2  - 4(3)(-168) > 0    NO

 

(3)  (-4)^2 -  4(1) (-4) >  0    NO

 

(4)  (30)^2 -4(25)(9) =  900 - 900  =  0    [ this is the one with the repeated root ]

 

 

 

cool cool cool

 Apr 7, 2020
 #3
avatar+932 
+2

floccinaucini, if you didn't know, when written in the form ax^2+bx+c, the discriminant of the quadratic equation is b^2-4ac.

 

Hope it helps!

HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #4
avatar+50 
+3

Repeated roots occur when the quadratic can be factored into two or more binomials. 

For example....

(x^2 + 6x + 9) has repeated roots because it can be factored into (x + 3)(x + 3), which only give one distinct root which is -3. 

 

On the other hand...

(x^2 + 5x + 6) does not have repeated roots because it cannot be factored into a product of two of the same binomials and factors to (x + 2)(x + 3), which gives us too distinct roots which are -2, and -3. 

 

The first one doesn't factor evenly. 

The second one factor into 3(x-8)(x+7), which has two distinct roots which are 8, and -7. 

The third one doesn't factor evenly. 

The fourth one factors into (5x-3)(5x-3) which has repeating roots which are 3/5 and is the answer. 

The fifth one factors into (x-12)(x+2) which has two distinct roots which are 12 and -2. 

 

Therefore, our answer is the 4th quadratic which is 25x^2 - 30x + 9, which factorizes (5x-3)^2 and has one repeating root that is 3/5. 

 Apr 7, 2020

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