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# One more question @Voldemort! Please do a full walkthrough; thank you!

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

Jul 23, 2022

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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
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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
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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)   Jul 23, 2022