Find the number of values of x for which 8x-27x-12x-18x-7-6

The correct option is A 2 Let 2^x=a and 3^x=b, the equation becomes a^3+b^3/a^2b+b^2a=7/6 or a^2 ab+b^2/ab=7/6 or 6a^2 13ab+6b^2=0 (2a 3b)(3a 2b)=0 There ...

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Quantitative Aptitude

Equations

Find the numb...

Question

Find the number of values of x for which $\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$

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Solution

The correct option is B
2

Let $2^{x} = a$ and $3^{x} = b,$ the equation becomes

$\frac{a^{3} + b^{3}}{a^{2 b} + b^{2 a}} = \frac{7}{6}$

$o r \frac{a^{2} - a b + b^{2}}{a b} = \frac{7}{6}$

$o r 6 a^{2} - 13 a b + 6 b^{2} = 0$

$(2 a - 3 b) (3 a - 2 b) = 0$

Therefore $2^{x + 1} = 3^{x + 1}$ or $2^{x - 1} = 3^{x - 1}$ which implies that x= -1 and x= 1. Hence only two solutions exist.

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Find the number of values of x for which 8x+27x12x+18x=76

The correct option is B 2 Let 2x=a and 3x=b, the equation becomes a3+b3a2b+b2a=76 or a2−ab+b2ab=76 or 6a2−13ab+6b2=0 (2a−3b)(3a−2b)=0 Therefore 2x+1=3x+1 or 2x−1=3x−1 which implies that x= -1 and x= 1. Hence only two solutions exist.

Find the number of values of x for which $\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$