Final answer:
To show that 0x3+ex=0 has one real root, use the Intermediate Value Theorem, which proves a root exists when a function changes sign over an interval. Unlike quadratic equations that use the quadratic formula to find roots, this equation involves e^x and requires understanding of exponential functions. Two-Dimensional Graphing and understanding how to solve for unknowns in equations are key skills in determining roots.
Step-by-step explanation:
The equation 0x3+ex=0 is not a quadratic equation, but you can still show that it has exactly one real root using calculus concepts such as the Intermediate Value Theorem or other root-finding methods.
Since the equation involves an exponential function e^x, it is transcendental rather than polynomial. To prove that there is exactly one real root, you can show that the function changes sign over an interval, which indicates by the Intermediate Value Theorem that a root must exist within that interval.
Moreover, the nature of the exponential function suggests that as x approaches negative infinity, e^x approaches zero, and as x approaches positive infinity, e^x becomes increasingly large, thus exhibiting only one point of intersection with the x-axis, which corresponds to the single real root.
For quadratic equations of the form ax² + bx + c = 0, roots can be found using the quadratic formula. In some cases, such as when constructing quadratic equations based on physical data, the roots have real values, and positive roots are often significant.
When graphing functions to find roots, Two-Dimensional (x-y) Graphing is a valuable method for visualization. Furthermore, in equilibrium problems and other math applications, it's important to know how to calculate various types of roots, including square roots and cube roots, using a calculator or other computational tools.
If the equation contained only one unknown and all the other values were known, rearranging the equation to solve for x would be a straightforward process. This is not the case with the original equation, since it involves an exponential function of x, but comprehending how to manipulate equations is a critical skill in mathematics.