Question

What is :

(49^x)(7^x^2)=2401^2

Answer 1

`(49^x)(7^(x^2))=2401^2\\ \\7^(2x) \cdot7^(x^2)=(7^4)^2\\ \\7^(2x+x^2)=7^8\ \ \ \Leftrightarrow\ \ \ 2x+x^2=8\ \ \ \Leftrightarrow\ \ \ x^2+2x-8=0\\ \\x^2-2x+4x-8=0\ \ \ \Leftrightarrow\ \ \ x(x-2)+4(x-2)=0\\ \\(x-2)(x+4)=0\ \ \ \Leftrightarrow\ \ \ ( x-2=0\ \ \ \vee\ \ \ x+4=0)\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=-4`

Answer 2

`(49^x)(7 ^(x^2))=2401^2 \\ \\( (7^2)^x)(7 ^(x^2))= (7^4) ^2 \\ \\ 7 ^(2x)\cdot 7^(x^2)=7^(8) \\ \\ 7 ^(2x+x^2) =7^(8) \\ \\2x+x^2 =8`


`x^2-2x-8 = 0\\ \\ a=1, \ b=-2, \ c=-8 \\ \\\Delta =b^2-4ac = (-2)^2 -4\cdot1\cdot (-8) = 4 +32 =36 \\ \\x_(1)=(-b-√(\Delta) )/(2a)=(2-√(36))/(2 )=( 2-6)/(2)=(-4)/(2)=-2 \\ \\x_(2)=(-b+√(\Delta) )/(2a)=(2+√(36))/(2 )=( 2+6)/(2)=(8)/(2)= 4 \\ \\Answer: \ x=-2 \ \ or \ \ x=4`