# Determining the Roots of an Equation with Two Real Solutions

• Hollysmoke
In summary, the conversation discusses two questions. The first question is about a fundamental problem in crystallography, specifically the determination of the packing fraction of a crystal lattice. The formula for calculating the packing fraction when the lattice contains two different kinds of atoms is also mentioned. The second question is about inputting a formula into Maple to differentiate it. The conversation also briefly mentions trying to use the Mean Value Theorem to prove the existence of two real roots for a given equation, but the speaker is seeking further help with the differentiation question.
Hollysmoke
I just have two questions:

A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown that the packing fraction is given by the formula:

f(x) = K(1+c^2x^3)/(1+x)^3

where x=r/R is the ratio of the radii, r and R of the two kinds of =atoms in the lattice, and c and K are positive constants.

How can I input this into maple to differentiate it?

And also:

x^4+3x^3-2=0

I'm supposed to prove that there are exactly 2 real roots. I tried using MVT but I'm not getting the answer =(

I figured out the 2nd one ^^ Just need help with the differentiation one.

To input the equation into Maple, you can use the "diff" function to differentiate it. The syntax for this would be "diff(f(x),x)" where f(x) is your equation. This will give you the derivative of the equation.

To prove that the equation x^4+3x^3-2=0 has exactly 2 real roots, you can use the Intermediate Value Theorem (IVT). The IVT states that if a continuous function f(x) has values of opposite signs at points a and b, then there must be at least one root between a and b. In this case, you can choose two points, one with a positive value and one with a negative value, and show that the function has values of opposite signs at those points. This would prove that there are at least two real roots. To show that there are exactly two roots, you would also need to show that there are no other real roots besides the two you have already found. This can be done by showing that the function does not change signs again between the two roots that you have found.

## What is the process for determining the roots of an equation with two real solutions?

The process for determining the roots of an equation with two real solutions involves setting the equation equal to 0 and then using methods such as factoring, the quadratic formula, or graphing to find the values of x that make the equation true. These values of x are the roots of the equation.

## How do you know if an equation has two real solutions?

An equation will have two real solutions if the discriminant, which is the value under the square root in the quadratic formula, is positive. This indicates that there are two distinct values of x that will make the equation true.

## What is the significance of having two real solutions for an equation?

Having two real solutions for an equation means that there are two distinct points where the equation crosses the x-axis on a graph. This can be useful in solving real-world problems, as the two solutions represent two possible answers to the equation.

## Can an equation have more than two real solutions?

Yes, an equation can have more than two real solutions. This is possible when the degree of the equation is higher than 2, such as in a cubic or quartic equation. In these cases, there can be three or four real solutions respectively.

## How can you verify that the solutions obtained for an equation are correct?

To verify that the solutions obtained for an equation are correct, you can substitute them back into the original equation and see if it results in a true statement. If it does, then the solutions are correct. Additionally, you can use a graphing calculator or software to graph the equation and see if the solutions align with the points where the graph crosses the x-axis.

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