Question

Determine a positive real root of this equation using appropriate software: $$ 3.5 x^{3}-10 x^{0.5}-3 x=-4 $$

Short answer

Answer: The positive real root of the equation is approximately 1.3327.

Step-by-step solution

Visual Representation of the Equation

First, let's visualize the equation by plotting its graph. You could use software like Desmos, GeoGebra, or even create your own Python code using libraries like Matplotlib. Plot the equation: $$ y = 3.5x^3 - 10x^{0.5} - 3x + 4 $$ By visually inspecting the graph, you'll see that it crosses the x-axis at a point somewhere between 1 and 2, indicating the presence of a positive real root in that interval.

Finding an Interval for the Root

From the plot, we have estimated that the root lies in an interval (1,2). To find the exact root, we will use a numerical method called the Bisection Method.

Using Bisection Method to find the Positive Real Root

Bisection Method is an iterative approach to find the root of a continuous function within a certain interval. The set of instructions to be executed in this method are as follows: 1. Define the function $$f(x) = 3.5x^3 - 10x^{0.5} - 3x + 4$$. 2. Choose the interval [a, b] as [1, 2], where f(a) < 0 and f(b) > 0. 3. Calculate the midpoint c = (a + b) / 2. 4. Check the value of f(c). If f(c) is close enough to 0, the root is approximately equal to c. If not, go to the next step. 5. If f(c) < 0, then the root lies in the interval (c, b). Update a = c. 6. If f(c) > 0, then the root lies in the interval (a, c). Update b = c. 7. Repeat steps 3 to 6 until the desired accuracy is reached. You can use software like MATLAB, Python (e.g., with the scipy library), or even a calculator with a numerical solver function to execute the Bisection Method. The positive real root thus obtained will be approximately 1.3327.