Final answer:
The solutions of the equation -2x⁴+16x²+32=3x⁴-2x³ are x = (8 ± √216) / 10.
Step-by-step explanation:
To find the solutions of the equation -2x⁴+16x²+32=3x⁴-2x³, we need to set the equation equal to zero and simplify it. Rearranging the equation, we get 5x⁴ - 2x³ - 16x² - 32 = 0. This can be factored to (x² + 4)(5x² - 8x - 8) = 0. Setting each factor equal to zero, we get two possible solutions: x² + 4 = 0 and 5x² - 8x - 8 = 0.
For x² + 4 = 0, we subtract 4 from both sides and take the square root to get x = ±√(-4). Since we can't take the square root of a negative number, this equation has no real solutions.
For 5x² - 8x - 8 = 0, we can use the quadratic formula to solve for x. Plugging in the values a = 5, b = -8, and c = -8 into the quadratic formula, we get x = (8 ± √(8² - 4(5)(-8))) / (2(5)). Simplifying this expression, we get two solutions: x = (8 ± √216) / 10.