X, Y and Z are long!

Algebra Level 4

Let P = [ ( x − y 2 y − x − x 2 + y 2 + y − 2 x 2 − x y − 2 y 2 ) ÷ 4 x 4 + 4 x 2 y + y 2 − 4 x 2 + y + x y + x ] ÷ x + 1 2 x 2 + y + 2 P=\left[\left(\frac{x-y}{2y-x} -\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\right) \div \frac{4x^4+4x^2y+y^2-4}{x^2+y+xy+x}\right] \div \frac{x+1}{2x^2+y+2}

Replace x = − 1.76 \displaystyle x=-1.76 and y = 3 25 \displaystyle y=\frac{3}{25} , what is P P equals to?


The answer is 0.5.

Adam Phúc Nguyễn by Adam Phúc Nguyễn , 20, Vietnam

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1 solution

I will show the main steps of factoring P P

P = [ ( x − y 2 y − x − x 2 + y 2 + y − 2 x 2 − x y − 2 y 2 ) ÷ 4 x 4 + 4 x 2 y + y 2 − 4 x 2 + y + x y + x ] ÷ x + 1 2 x 2 + y + 2 P=\left[\left(\frac{x-y}{2y-x} -\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\right) \div \frac{4x^4+4x^2y+y^2-4}{x^2+y+xy+x}\right] \div \frac{x+1}{2x^2+y+2}

P = [ ( x − y ) ( x + y ) + x 2 + y 2 + y − 2 ( x + y ) ( 2 y − x ) × ( x + y ) ( x + 1 ) ( 2 x 2 + y + 2 ) ( 2 x 2 + y − 2 ) ] × 2 x 2 + y + 2 x + 1 P=\left[\frac{(x-y)(x+y)+x^2+y^2+y-2}{(x+y)(2y-x)} \times \frac{(x+y)(x+1)}{(2x^2+y+2)(2x^2+y-2)}\right] \times \frac{2x^2+y+2}{x+1}

P = ( 2 x 2 + y − 2 2 y − x × x + 1 2 x 6 2 + y − 2 ) × 1 x + 1 = 1 2 y − x P=\left(\frac{2x^2+y-2}{2y-x} \times \frac{x+1}{2x6^2+y-2}\right) \times \frac{1}{x+1} =\frac{1}{2y-x}

Replace x = − 1.76 \displaystyle x=-1.76 and y = 3 25 \displaystyle y=\frac{3}{25} to P P , the final answer of P P is 1 2 \boxed{\frac{1}{2}}

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