x can take finite values!

Algebra Level 3

For a real x x satisfying 6 x 2 − 5 x − 3 x 2 − 2 x + 6 ≤ 4 , \dfrac{6x^2-5x-3}{x^2-2x+6} \leq 4 ,

the least and the highest values of 4 x 2 4x^2 are A A and B B respectively.

Find the value of A − B A-B .


The answer is -81.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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2 solutions

Ayush Verma
Nov 6, 2014

I think,This problem has been reported by multiple members because they are subtracting LEAST from the HIGHEST and getting +81 instead of -81

x 2 − 2 x + 6 = ( x − 1 ) 2 + 5 { x }^{ 2 }-2x+6={ (x-1) }^{ 2 }+5 is always +ve so multiply by it both side without affecting inequality sign.

6 x 2 − 5 x − 3 ≤ 4 x 2 − 8 x + 24 ⇒ 2 x 2 + 3 x − 27 ≤ 0 ⇒ ( 2 x + 9 ) ( x − 3 ) ≤ 0 ⇒ x ∈ [ − 9 2 , 3 ] ⇒ x 2 ∈ [ 0 , 81 4 ] ⇒ A − B = 4 ( 0 − 81 4 ) = − 81 6{ x }^{ 2 }{ -5x-3\le 4x }^{ 2 }-8x+24\\ \\ \Rightarrow 2{ x }^{ 2 }+3x-27\le 0\\ \\ \Rightarrow (2x+9)(x-3)\le 0\\ \\ \Rightarrow x\in \left[ \cfrac { -9 }{ 2 } ,3 \right] \quad \Rightarrow { x }^{ 2 }\in \left[ 0,\cfrac { 81 }{ 4 } \right] \\ \\ \Rightarrow A-B=4\left( 0-\cfrac { 81 }{ 4 } \right) =-81

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Nice solution.

Sandeep Bhardwaj - 6 years, 7 months ago

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Thanks,sir

Ayush Verma - 6 years, 7 months ago

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Before disputing any problem, you first need to be 100% sure. Otherwise none should report the problem. And must look at the solution (if given) before going to report any problem. @brilliant

Sandeep Bhardwaj - 6 years, 7 months ago

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@Sandeep Bhardwaj – And you should have seen my face when at first 81 81 was marked wrong.

Satvik Golechha - 6 years, 4 months ago

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@Satvik Golechha – Hahaha.Easy problem LOL

Gautam Sharma - 6 years, 3 months ago

@Satvik Golechha – waiting for you march newsletter.

Gautam Sharma - 6 years, 3 months ago

@Sandeep Bhardwaj – Yes i agree.

Ayush Verma - 6 years, 7 months ago

Same way as I did.

Aayush Patni - 6 years, 4 months ago

I did 81 - 36 first and later realised while solving again that it's minimum value will be zero! Nice solution !😀

Anurag Pandey - 4 years, 10 months ago
Afreen Sheikh
Jan 6, 2015

the value of x after solving the inequalities is [-4.5,3] but after squaring the value of x^2 is [0,20.25] hence the least A=0 and highest B=4x20.25=81 and 0-81=-81 hence the solution

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