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Let \(a\) and \(b\) be the solutions of the equation \(2x^2-10x+5=0\). What is the value of \((2a-3)(4b-6)\)?

 Jul 12, 2022
 #1
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The solutions to 2x^2 - 10x + 5 = 0 are (5 +/- sqrt(15))/2  Plugging these values in, you get

(2*(5 + sqrt(15))/2 - 3)(4*(5 - sqrt(15))/2 - 6) = -18

 Jul 12, 2022
 #2
avatar+124525 
+1

(2a - 3) ( 4b - 6)  =

 

(2a - 3) (2) (2b - 3)  =

 

2 ( 2a - 3) (2b - 3)  =

 

2 [ 4ab - 6a - 6b + 9 ]

 

2 [ 4ab  - 6 ( a+ b) + 9 ] 

 

The product of the roots = 5/2

The sum of the roots = - (-10) / 2 =  5

 

So

 

2 [  4 (5/2)  - 6 ( 5)  + 9 ]  =

 

2 [ 10 - 30 + 9 ]  =

 

2 [ -11 ]  = 

 

-22

 

 

cool cool cool

 Jul 12, 2022
 #3
avatar+195 
0

Another solution! :D

 

P(x) = 2x^2 - 10x + 5 = 2(x - a)(x - b)

 

(2a - 3)(4b - 6) = 8(a - 3/2)(b - 3/2)

 

P(3/2) = 2(3/2 - a)(3/2 - b) = -11/2

 

-11/2 * 4 = -22. 

 Jul 13, 2022
edited by Voldemort  Jul 13, 2022

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