Complicated Expression

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Let x 1 x_1 and x 2 x_2 be the roots of the equation x 2 − 3 x + 1 = 0 x^2-3x+1=0 . The value of:

x 1 x 2 + x 2 x 1 \frac{x_1}{\sqrt{x_2}}+\frac{x_2}{\sqrt{x_1}} can be written as a b a\sqrt{b} where a a and b b are coprime positive integers.What is a + b a+b ?


The answer is 7.

Lorenc Bushi by Lorenc Bushi , 21, Albania

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1 solution

x 2 − 3 x + 1 = 0 , x 1 ≥ x 2 x^2-3x+1=0, \; x_1 \geq x_2 Bhaskara’s Formula \text{Bhaskara's Formula} x 1 = 3 + 5 2 x 2 = 3 − 5 2 x_1 = \dfrac{3 + \sqrt{5}}{2} \; \; \; \; \; x_2 = \dfrac{3 - \sqrt{5}}{2} Golden Ratio Identity \text{Golden Ratio Identity} x 1 = 5 + 1 2 x 2 = 5 − 1 2 \sqrt{x_1} = \dfrac{\sqrt{5} + 1}{2} \; \; \; \; \; \sqrt{x_2} = \dfrac{\sqrt{5} - 1}{2} a b = x 1 x 2 + x 2 x 1 ⇒ a b ( x 1 − x 1 x 2 + x 2 ) ( x 1 + x 2 ) x 1 x 2 a\sqrt{b} = \dfrac{x_1}{\sqrt{x_2}} + \dfrac{x_2}{\sqrt{x_1}} \Rightarrow a\sqrt{b} \dfrac{(x_1 - \sqrt{x_1 x_2} + x_2)(\sqrt{x_1} + \sqrt{x_2})}{\sqrt{x_1 x_2}} Vieta’s Formulae \text{Vieta's Formulae} a b = ( 3 − 1 ) 5 1 a\sqrt{b} = \dfrac{(3 - 1) \sqrt{5}}{1} a b = 2 5 a\sqrt{b} = 2\sqrt{5} a + b = 7. \boxed{a+b=7.}

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Guilherme, here is another solution for those who are not familiar with some of your methods:

If we square the given expression,we will obtain:

( x 1 x 2 + x 2 x 1 ) 2 = x 1 2 x 2 + x 2 2 x 1 + 2 x 1 x 2 x 1 x 2 = x 1 3 + x 2 3 x 1 x 2 + 2 x 1 x 2 x 1 x 2 (\frac{x_1}{\sqrt{x_2}}+\frac{x_2}{\sqrt{x_1}})^2=\frac{x_1^2}{x_2}+\frac{x_2^2}{x_1}+\frac{2x_1x_2}{\sqrt{x_1x_2}}=\frac{x_1^3+x_2^3}{x_1x_2}+\frac{2x_1x_2}{\sqrt{x_1x_2}} .Observe:

x 1 3 + x 2 3 = ( x 1 + x 2 ) 3 − 3 x 1 x 2 ( x 1 + x 2 ) x_1^3+x_2^3=(x_1+x_2)^3-3x_1x_2(x_1+x_2) .From vietas formulas we have :

x 1 + x 2 = 3 x_1+x_2=3 ( 1 ) (1)

x 1 x 2 = 1 x_1x_2=1 ( 2 ) (2)

Therefore x 1 3 + x 2 3 = 18 x_1^3+x_2^3=18 and the expression is equal to :

18 1 + 2 ∗ 1 1 \frac{18}{1}+\frac{2*1}{\sqrt{1}} and by making the operations we obtain :

( x 1 x 2 + x 2 x 1 ) 2 = 20 (\frac{x_1}{\sqrt{x_2}}+\frac{x_2}{\sqrt{x_1}})^2=20 where :

x 1 x 2 + x 2 x 1 = 20 = 2 5 \frac{x_1}{\sqrt{x_2}}+\frac{x_2}{\sqrt{x_1}}=\sqrt{20}=2\sqrt{5} .Thus, a = 2 a=2 and b = 5 b=5 and a + b = 7 \boxed{a+b=7}

Lorenc Bushi - 7 years, 5 months ago

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Way better than my solution. I got the easy job because the roots are powers oh φ \varphi .

Guilherme Dela Corte - 7 years, 5 months ago

Guilherme ,what if we had a similar problem to this one but with different equation ,would you still be able to use the Golden Ratio identity ?

Lorenc Bushi - 7 years, 5 months ago

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I would not. This only works because x 1 x_1 and x 2 x_2 are powers of φ \varphi , where φ 2 = φ + 1 \varphi^2 = \varphi + 1 .

Guilherme Dela Corte - 7 years, 5 months ago

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