4 α + 4 β 4^{\alpha}+4^{\beta}

Algebra Level 3

Let α \alpha and β \beta be the two roots of the equation 2 2 x − 13 ⋅ 2 x + 1 = 0. 2^{2x}-13 \cdot 2^x+1=0. What is the value of 4 α + 4 β ? 4^{\alpha}+4^{\beta}?

167 161 165 163

Ajay Dutta by Ajay Dutta , 24, India

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2 solutions

( 2 x ) 2 − 13 ⋅ 2 x + 1 = 0 (2^x)^2 - 13 \cdot 2^x + 1 = 0 2 a + 2 b = 13 , 2 a ⋅ 2 b = 1 2^a + 2^b = 13, \; 2^a \cdot 2^b = 1 ( 2 a ) 2 + ( 2 b ) 2 = ( 2 a + 2 b ) 2 − 2 ⋅ 2 a ⋅ 2 b (2^a)^2 + (2^b)^2 = (2^a + 2^b )^2 - 2 \cdot 2^a \cdot 2^b 4 a + 4 b = 167 \boxed{4^a + 4^b = 167}

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Moshiur Mission
Mar 30, 2014

22x-13(2x)+1=0 If solution for x is α, β then 2α+2β=13 and 2α2β=1 Now 4α+4β= (2α)2+(2β)2=(2α+2β)2-2.2α2β=132-2.1=169-2=167

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