Completing the square

Algebra Level 3

If 2 x 2 + y 2 = 6 x 2x^2+y^2=6x , where x , y x,y are reals, find the maximum of x 2 + y 2 + 2 x x^2+y^2+2x .

16 15 12 4 Maximum does not exist

Dexter Woo Teng Koon by Dexter Woo Teng Koon , 20, Malaysia

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1 solution

From 2 x 2 + y 2 = 6 x 2x^2+y^2=6x

We have 6 x − 2 x 2 ≥ 0 6x-2x^2\ge 0

⟹ 0 ≤ x ≤ 3 \implies 0\le x\le 3

x 2 + y 2 + 2 x = x 2 + 6 x − 2 x 2 + 2 x = 16 − ( x − 4 ) 2 ≤ 16 − ( 3 − 4 ) 2 = 15 x^2+y^2+2x=x^2+6x-2x^2+2x=16-(x-4)^2\le 16-(3-4)^2=15

5 Helpful 0 Interesting 0 Brilliant 0 Confused

It'd be much nicer if you put the condition that x , y x,~y are all real numbers. ;)

Boi (보이) - 3 years, 11 months ago

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Edited, thanks :)

Dexter Woo Teng Koon - 3 years, 11 months ago

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