Question 7
The number of real numbers $$x$$ which satisfy $$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$ is
Solution
Put $$a=2^x$$ and $$b=3^x$$. The equation becomes $$\frac{a^3+b^3}{a^2b+ab^2}=\frac{7}{6}$$, which simplifies to $$6a^2-13ab+6b^2=0$$. Hence $$(3a-2b)(2a-3b)=0$$, giving $$\left(\frac{2}{3}\right)^x=\frac{2}{3}$$ or $$\left(\frac{2}{3}\right)^x=\frac{3}{2}$$. Therefore $$x=1$$ or $$x=-1$$, so there are $$2$$ real solutions.
Get AI Help
Book Free CAT Mentorship
Get personalized CAT strategy from a 99%iler
500+ students mentored
OTP Verification
Enter the 6-digit code sent to your phone
Booking Summary
Session:Free CAT Mentorship
Phone:
Date:
Time:
Enter OTP
Didn't receive the OTP?
