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If $(x+2)(x-3)=14$, find the sum of the possible values of $x$.

 Jul 23, 2022
 #1
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Oh, Sorry. one more question; i got the last one:

 

Find $(1)/(a-1)+(1)/(b-1),$ where $a$ and $b$ are the roots of the quadratic equation $2x^2-7x+2 = 0.$

 

Thanks! :)

 Jul 23, 2022
 #3
avatar+2322 
+1

Note that \({1 \over a - 1} + {1 \over b - 1} = {b-1 \over (a - 1)(b-1)} + {a-1 \over (a - 1)(b-1)} = {a + b - 2 \over ab−a−b+1} = {a + b - 2 \over ab−(a+b)+1} \)

 

By Vieta's, \(a + b = -{b \over a} = {7 \over 2}\) and \(ab = {c \over a} = {2 \over 2} = 1 \)

 

Now, just plug these back into the original equation and simplify...

BuilderBoi  Jul 24, 2022
 #2
avatar+124524 
+1

2x^2 - 7x + 2  = 0

 

The sum of the roots =  7/2

The product of the roots  =  2/2  = 1

 

1/ (a - 1)  + 1/ (b -1)  =

 

(a - 1) + (b -1)

___________  =

(a - 1) (b -1)

 

(a + b) - 2

____________   =

ab - (a + b) + 1

 

(7/2) - 2

____________   =

1 - (7/2) + 1

 

  (3/2)

______  =      -1

 - (3/2)

 

 

cool cool cool

 Jul 23, 2022

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