Algebraic Geometry?

Algebra Level 4

Find values of m m for which the expression 2 x 2 + m x y + 3 y 2 − 5 y − 2 2{x}^{2} + mxy + 3{y}^{2} -5y-2 can be factorized into two linear factors.

If the values are α \alpha and β \beta , find ∣ α ∣ + ∣ β ∣ |\alpha| + |\beta| .


The answer is 14.

Swapnil Das by Swapnil Das , 20, India

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1 solution

Rohit Ner
Jun 1, 2016

For the given expression to be a product of two linear factors, it must represent a pair of lines.

Equation of a pair of lines is given by a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0 where ∣ a h g h b f g f c ∣ = 0 \left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} \right| =0

Plugging in the corresponding values, we get

∣ 2 m 2 0 m 2 3 − 5 2 0 − 5 2 − 2 ∣ = 0 m 2 = 49 ⇒ m = 7 , − 7 ∣ α ∣ + ∣ β ∣ = ∣ 7 ∣ + ∣ − 7 ∣ = 14 \begin{aligned} \left| \begin{matrix} 2 & \cfrac { m }{ 2 } & 0 \\ \cfrac { m }{ 2 } & 3 & -\cfrac { 5 }{ 2 } \\ 0 & -\cfrac { 5 }{ 2 } & -2 \end{matrix} \right|& =0\\{m}^2&=49\\\Rightarrow m&=7,-7\\|\alpha| + |\beta|&=|7| + |-7|\\&\huge\color{#3D99F6}{=\boxed{14}}\end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

@Swapnil Das Why the title?Did you use diff.?

Anik Mandal - 5 years ago

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I was trying to confuse the solvers, because higher powers and expressions are good signs of differentiation problems :P

Swapnil Das - 5 years ago

@Rohit Ner Excellent and beautiful solution.

Swapnil Das - 5 years ago

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