Final answer:
To find the smaller solution of the exponential equation 6ˣ²=36¹/²x+3), we need to simplify the equation and solve for x. This is a quadratic equation, and by using the quadratic formula, we can find the two possible solutions. The smaller solution is x = -0.363.
Step-by-step explanation:
To find the smaller solution of the exponential equation 6ˣ²=36¹/²x+3), we need to simplify the equation and solve for x. We can start by writing the equation in a similar form with the same exponents. Let's multiply both sides of the equation by 2 to eliminate the fraction: 12ˣ²=72¹/²x+6. Rearranging the equation, we get: 12ˣ² - 72¹/²x - 6 = 0.
This is a quadratic equation, so we can solve it using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a), where a = 12, b = -72¹/², and c = -6. Plugging in these values, we can solve for x.
Using the quadratic formula, we find two possible solutions: x = -0.363 or x = 6.639. Since we are looking for the smaller solution, the answer is x = -0.363.