Welcome 2016! Part 8

Algebra Level 2

1 1 3 + 5 3 1 1 2 + x + 5 2 = 16 \large \dfrac{11^3 +5^3}{11^2+ x +5^2}=16

Find the value of x x .

-110 -55 110 55

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Anish Harsha by Anish Harsha , 20, India

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4 solutions

Let x = 11 \color{#D61F06}{x}=11 and y = 5 \color{#3D99F6}{y}=5 .
⇒ x 3 + y 3 x 2 + y 2 + □ = 16 \Rightarrow \dfrac{\color{#D61F06}{x^3}+\color{#3D99F6}{y^3}}{\color{#D61F06}{x^2}+\color{#3D99F6}{y^2}+\color{#624F41}{\square}}=16

= ( x + y ) ( x 2 + y 2 − x y ) = 16 x 2 + 16 y 2 + 16 □ =(\color{#D61F06}{x}+\color{#3D99F6}{y})(\color{#D61F06}{x^2}+\color{#3D99F6}{y^2}-\color{#D61F06}{x}\color{#3D99F6}{y})=16\color{#D61F06}{x^2}+16\color{#3D99F6}{y^2}+16\color{#624F41}{\square}
= ( 16 ) ( x 2 + y 2 − x y ) = 16 ( x 2 + y 2 + □ ) =(16)(\color{#D61F06}{x^2}+\color{#3D99F6}{y^2}-\color{#D61F06}{x}\color{#3D99F6}{y})=16(\color{#D61F06}{x^2}+y\color{#3D99F6}{^2}+\color{#624F41}{\square)}
= x 2 + y 2 − x y = x 2 + y 2 + □ =\color{#D61F06}{x^2}+\color{#3D99F6}{y^2}-\color{#D61F06}{x}\color{#3D99F6}{y}=\color{#D61F06}{x^2}+\color{#3D99F6}{y^2}+\color{#624F41}{\square}
⇒ □ = ( − x y ) = ( − 11 ) × 5 = − 55 \Rightarrow \color{#624F41}{\square}=(\color{#D61F06}{-x}\color{#3D99F6}{y})=(-11)×5=\boxed{-55}

9 Helpful 0 Interesting 0 Brilliant 0 Confused
Rohit Udaiwal
Dec 12, 2015

( 11 + 5 ) ( 1 1 2 − 55 + 5 2 ) 1 1 2 + □ + 5 2 = 16 ⟹ 1 1 2 − 55 + 5 2 = 1 1 2 + □ + 5 2 ∴ □ = − 55. \dfrac{(11+5)(11^2-55+5^2)}{11^2+\square+5^2}=16 \implies 11^2-55+5^2=11^2+\square+5^2 \therefore \square=-55.

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Vishesh Jha
Dec 13, 2015

Intelligence

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I simply substitute the choices . I ain't that genius.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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