Question

Determine the roots of the function f(x) = (5^(2x) − 6)^2 − (5^(2x) − 6) − 12

Answer 1

The roots of the function f(x) are all values of x for which f(x)=0

$(5^{2x}-6)² -(5^{2x}-6)-12=0\\ (5^{2x}-6)²+3(5^{2x}-6)-4(5^{2x}-6)-12=0\\ (5^{2x}-6)((5^{2x}-6)+3)-4((5^{2x}-6)+3)=0\\ (5^{2x}-6)(5^{2x}-3)-4(5^{2x}-3)=0\\ (5^{2x}-3)(5^{2x}-6-4)=0\\ (5^{2x}-3)(5^{2x}-10)=0\\ (5^{2x}-3)=0 \textsf{ or }(5^{2x}-10)=0\\ 5^{2x}=3 \textsf{ or } 5^{2x}=10\\ \log_5{5^{2x}}=\log_5{3} \textsf{ or } \log_5{5^{2x}}=\log_5{10}\\ 2x=\log_5{3} \textsf{ or } 2x=\log_5{10}\\ x=\frac{1}2\log_5{3} \textsf{ or } x=\frac{1}2\log_5{10}\\ x=\log_5{3}^{\frac{1}2} \textsf{ or } x=\log_5{10}^{\frac{1}2}\\ x=\log_5\sqrt{3} \textsf{ or } x= \log_5\sqrt{10}$