What is:

\[
(49^x)(7^{x^2}) = 2401^2
\]

( 4 9 x ) ( 7 x 2 ) = 240 1 2 7 2 x ⋅ 7 x 2 = ( 7 4 ) 2 7 2 x + x 2 = 7 8 ⇔ 2 x + x 2 = 8 ⇔ x 2 + 2 x − 8 = 0 x 2 − 2 x + 4 x − 8 = 0 ⇔ x ( x − 2 ) + 4 ( x − 2 ) = 0 ( x − 2 ) ( x + 4 ) = 0 ⇔ ( x − 2 = 0 ∨ x + 4 = 0 ) . x = 2 x = − 4

Answered by kate200468 | 2024-06-10

( 4 9 x ) ( 7 x 2 ) = 240 1 2 (( 7 2 ) x ) ( 7 x 2 ) = ( 7 4 ) 2 7 2 x ⋅ 7 x 2 = 7 8 7 2 x + x 2 = 7 8 2 x + x 2 = 8
x 2 − 2 x − 8 = 0 a = 1 , b = − 2 , c = − 8 Δ = b 2 − 4 a c = ( − 2 ) 2 − 4 ⋅ 1 ⋅ ( − 8 ) = 4 + 32 = 36 x 1 ​ = 2 a − b − Δ ​ ​ = 2 2 − 36 ​ ​ = 2 2 − 6 ​ = 2 − 4 ​ = − 2 x 2 ​ = 2 a − b + Δ ​ ​ = 2 2 + 36 ​ ​ = 2 2 + 6 ​ = 2 8 ​ = 4 A n s w er : x = − 2 or x = 4

Answered by Lilith | 2024-06-10

The equation ( 4 9 x ) ( 7 x 2 ) = 240 1 2 can be rewritten in terms of powers of 7 . After simplifying, we arrive at the quadratic equation x 2 + 2 x − 8 = 0 , which factors to give solutions of x = 2 and x = − 4 .
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Answered by kate200468 | 2024-11-12