Solve over the Interval x^4+x-3=0 , (1,2)

Solution:

Step 1:

Plot the graph of the function for both sides of the equation. The x-coordinates where the graphs intersect are the solutions. We find that the solutions are approximately $x\approx -1.45262687$ and $x\approx 1.16403514$.

Step 2:

Determine which solutions fall within the given interval $(1,2)$.

Step 2.1:

Since the interval $(1,2)$ does not include $-1.45262687$, we discard this value. It is not a valid solution within the specified range.

Step 2.2:

The value $1.16403514$ lies within the interval $(1,2)$. Therefore, $x=1.16403514$ is the solution we are looking for within the given interval.

Knowledge Notes:

To solve the equation $x^4+x-3=0$ over a specific interval, such as $(1,2)$, we can use the following knowledge points:

1. Graphical Solution: By graphing the function, we can visually inspect where the function crosses the x-axis within the interval of interest. This gives us an approximate solution which can be further refined if necessary.

2. Intervals: An interval represents a range of values between two endpoints. In this case, the interval $(1,2)$ includes all real numbers greater than 1 and less than 2. It is important to check whether the solutions from the graph fall within this range.

3. Approximation: When we graph functions and look for intersections, we often get approximate values for the solutions. These approximations can be refined using more precise mathematical methods if required.

4. Discarding Irrelevant Solutions: Not all solutions to an equation may be relevant to the problem at hand. In this case, any solution outside the interval $(1,2)$ must be discarded as it does not satisfy the problem's constraints.

5. Equation Solving: The original equation $x^4+x-3=0$ is a polynomial equation. Polynomial equations can have multiple solutions, and the degree of the polynomial (in this case, 4) gives us the maximum number of real solutions we can expect.

6. LaTeX Formatting: When presenting mathematical solutions, LaTeX is used to format equations and mathematical expressions to make them clear and visually understandable. For instance, $x^4+x-3=0$ is rendered in LaTeX to display the equation properly.