Solve:
$18^{x^2+4 x+4}=18^{9 x+18}$
Answer (1)
Equate the exponents: x 2 + 4 x + 4 = 9 x + 18 .
Rearrange to quadratic form: x 2 − 5 x − 14 = 0 .
Factor the quadratic: ( x − 7 ) ( x + 2 ) = 0 .
Solve for x : x = − 2 and x = 7 . The solutions are x = − 2 and x = 7 .
Explanation
Equating the exponents We are given the equation 1 8 x 2 + 4 x + 4 = 1 8 9 x + 18 . Since the bases are equal, we can equate the exponents to solve for x .
Forming the equation Equating the exponents, we get: x 2 + 4 x + 4 = 9 x + 18
Rearranging to quadratic form Rearranging the terms to form a quadratic equation, we have: x 2 + 4 x + 4 − 9 x − 18 = 0
x 2 − 5 x − 14 = 0
Factoring the quadratic Now, we factor the quadratic equation: ( x − 7 ) ( x + 2 ) = 0
Solving for x Setting each factor to zero, we solve for x :
x − 7 = 0 ⇒ x = 7 x + 2 = 0 ⇒ x = − 2
Final solutions Therefore, the solutions are x = − 2 and x = 7 .
Examples
Imagine you are designing a bridge and need to calculate the stress on a support beam. The equation 1 8 x 2 + 4 x + 4 = 1 8 9 x + 18 is analogous to finding the points where two different stress functions are equal. Solving this equation helps engineers determine critical points where the stress is the same, ensuring the bridge's structural integrity. This type of problem demonstrates how solving exponential equations can be applied in real-world engineering scenarios to ensure safety and stability.
