This Should Be Simple, Right?

Algebra Level 4

p ( x ) = 51 x 2 + m x + c q ( x ) = 3 x 2 + b x + a \begin{aligned} p(x)&=&51x^2+mx+c \\ q(x)&=&3x^2+bx+a \end{aligned}

p ( x ) p(x) and q ( x ) q(x) are two quadratic polynomials with integer coefficients such that p ( r ) = q ( r ) = 0 p(r)=q(r)=0 . If r r is an irrational number, what is c a \dfrac{c}{a} ?


The answer is 17.

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Mursalin Habib by Mursalin Habib , 23, Bangladesh

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3 solutions

Since irrational roots occur in pairs, if the quadratics share one root, then they also share the other root. Thus, the co-efficients are in proportion.
51 3 = 17 = m b = c a \dfrac{51}{3} = 17 = \dfrac{m}{b} = \dfrac{c}{a}

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It'd be nice if you explained when irrational roots occur in pairs.

Do they occur in pairs for every polynomial? Do they occur in pairs whenever you have a quadratic polynomial?

Mursalin Habib - 5 years, 5 months ago

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They occur in pair if coefficients are rational.

Deepak Kumar - 5 years, 5 months ago

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Do they occur in pairs for every polynomial with rational coefficients?

Mursalin Habib - 5 years, 5 months ago

one can solve it with da help of limit applyin la hospitals rule dats da easiest.........

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I'm not sure I understand what you mean. Can you explain?

Mursalin Habib - 5 years ago
Abe Morillo
Dec 29, 2015

c/51 = a/3 c/a = 51/3 c/a = \box{17}

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