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One root of x^2 + 12x + k = 18x + 3 is twice the other root. Find k.

 Jul 5, 2021
 #1
avatar+22096 
+1

x2 + 12x + k  =  18x + 3     --->     x2 - 6x + k - 3

 

Using the quadratic formula with  a = 1,  b = -6,  c = k - 3:

 

x  =  [ - -6  +/-  sqrt( (-6)2 -  4(1)(k - 3) ) ]  /  2(1)

x  =  [ 6  +/-  sqrt( 36 - 4k + 12 ) ]  /  2

x  =  [ 6  +/-  2 sqrt( 12 - k ) ] /  2

x  =  3  +/-  sqrt( 12 - k )

 

The larger root (the "+" root) is twice as large as the smaller root (the "-" root):

 

3 + sqrt( k - 12 )  =  2(3 - sqrt( 12 - k )

3 + sqrt( 12 - k )  =  6 - 2 sqrt( 12 - k )

   3 sqrt( 12 - k )  =  3

      sqrt( 12 - k )  =  1

 [ sqrt( 12 - k ) ]2  =  (1)2

                 12 - k  =  1

                        k  =  11

 Jul 5, 2021
 #2
avatar+33376 
+1

x^2  -6x + k - 3 = 0

 

the two numbers added = -6          call them a and 2a       a + 2a = -6

                                                                                             a = - 2

     and          a * 2a = k-3

                        2a^2  = k-3

                        8 = k-3

                           k = 11

 Jul 5, 2021

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