It's a long equation

Algebra Level 4

If S S is the sum of all real x x satisfying

( 6 x − 3 ) 7 − 3 x + ( 15 − 6 x ) 3 x − 2 = 2 − 9 x 2 + 27 x − 14 + 11 , \left( 6x-3 \right) \sqrt { 7-3x } +\left( 15-6x \right) \sqrt { 3x-2 } =2\sqrt { -9{ x }^{ 2 }+27x-14 } +11,

what is the value of S S ?

7 3 4 \frac { 3 }{ 4 } 3 4

Linkin Duck by Linkin Duck , 33, Vietnam

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1 solution

Chew-Seong Cheong
Mar 26, 2017

( 6 x − 3 ) 7 − 3 x + ( 15 − 6 x ) 3 x − 2 = 2 − 9 x 2 + 27 x − 14 + 11 ( 6 x − 4 + 1 ) 7 − 3 x + ( 1 + 14 − 6 x ) 3 x − 2 = 3 x − 2 + 2 ( 3 x − 2 ) ( 7 − 3 x ) + 7 − 3 x + 6 Let a = 3 x − 2 , b = 7 − 3 x 2 a 2 b + b + a + 2 b 2 a = a 2 + 2 a b + b 2 + 6 Note that a 2 + b 2 = 5 2 a b ( a + b ) + a + b = ( a + b ) 2 + 6 ( ( a + b ) 2 − 5 ) ( a + b ) + a + b = ( a + b ) 2 + 6 ( a + b ) 3 − 4 ( a + b ) = ( a + b ) 2 + 6 ( a + b ) 3 − ( a + b ) 2 − 4 ( a + b ) − 6 = 0 ( ( a + b ) − 3 ) ( ( a + b ) 2 + 2 ( a + b ) + 2 ) = 0 ⟹ a + b = 3 ( a + b ) 2 = 9 5 + 2 a b = 9 2 a b = 4 a 2 b 2 = 4 − 9 x 2 + 27 x − 14 = 4 9 x 2 − 27 x + 18 = 0 x 2 − 3 x + 2 = 0 ( x − 1 ) ( x − 2 ) = 0 ⟹ x = { 1 2 \begin{aligned} (6x-3)\sqrt{7-3x} + (15-6x)\sqrt{3x-2} & = 2\sqrt{-9x^2+27x-14} + 11 \\ (6x-4+1)\sqrt{7-3x} + (1+14-6x)\sqrt{3x-2} & = 3x -2 + 2\sqrt{(3x-2)(7-3x)} +7-3x + 6 & \small \color{#3D99F6} \text{Let }a = \sqrt{3x-2}, \ b=\sqrt{7-3x} \\ 2a^2b + b + a + 2b^2a & = {\color{#3D99F6}a^2 + 2ab+b^2} + 6 & \small \color{#3D99F6} \text{Note that } a^2+b^2 = 5 \\ {\color{#3D99F6}2ab}(a+b) + a+b & = (a+b)^2 + 6 \\ {\color{#3D99F6}\left((a+b)^2 -5 \right)}(a+b) + a+b & = (a+b)^2 + 6 \\ (a+b)^3 - 4(a+b) & = (a+b)^2 + 6 \\ (a+b)^3 - (a+b)^2 - 4(a+b) - 6 & = 0 \\ \left((a+b)-3\right) \left( (a+b)^2+ 2(a+b)+2\right) & = 0 \\ \implies a+b & = 3 \\ (a+b)^2 & = 9 \\ 5+2ab & = 9 \\ 2ab & = 4 \\ a^2b^2 & = 4 \\ -9x^2+27x-14 & = 4 \\ 9x^2-27x +18 & = 0 \\ x^2-3x+2 & = 0 \\ (x-1)(x-2) & = 0 \\ \implies x & = \begin{cases} 1 \\ 2 \end{cases} \end{aligned}

⟹ S = 1 + 2 = 3 \implies S = 1+2=\boxed{3}

1 Helpful 0 Interesting 1 Brilliant 0 Confused

Is that a = 7 − 3 x , b = 3 x − 2 a=\sqrt { 7-3x } , b=\sqrt { 3x-2 } ? There might be a typo mistake on the line 2, Mr. Chew-Seong

Linkin Duck - 4 years, 2 months ago

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Thanks. They were typos.

Chew-Seong Cheong - 4 years, 2 months ago

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