Intersection of Graphs: Solving for C and D Using Equations

Introduction

In this article, we explore the intersection of the graphs of two functions: y -4x5 and y x3/2 - 3x - 1. The intersection point of these graphs yields the mutual x-coordinate, which is the solution to the equation x3/2 - Cx - D 0. The goal is to determine the values of C and D that satisfy this equation.

Methodology

The first step is to set the two equations equal to each other:

x3/2 - 3x - 1 -4x - 5

After simplifying the equation, we get:

x3/2 -x4

To solve for x, square both sides of the equation:

(x3/2)2 (-x4)2

Which results in:

x3 x8 - 8x4 16

Rearrange the equation to set it to zero:

x3 - x8 8x4 - 16 0

From there, solve for x:

x 1.7281632015

Solving for C and D

Now, to find the values of C and D, we consider the equation:

x3/2 - Cx - D 0

First, rearrange the equation to isolate x3/2:

x3/2 Cx D

Square both sides to eliminate the fractional exponent:

(x3/2)2 (Cx D)2

Which results in:

x3 C2x2 2CDx D2

Rearrange to set the equation to zero:

x3 - C2x2 - 2CDx - D2 0

By comparing the coefficients of the equations x3 - x8 8x4 - 16 0 and x3 - C2x2 - 2CDx - D2 0, we get:

-C2 -1

C2 1

Therefore, C -1 or 1

-D2 -16

D2 16

Therefore, D -4 or 4

To satisfy the condition that the product of C and D be -4, we have:

C * D -4

Thus, the solutions are:

C -1, D 4

and

C 1, D -4

Conclusion

From the graph and the solutions derived above, we conclude that the values of C and D that satisfy the equation are:

C 1, D -4

This method allows us to determine the intersection point of the two graphs using algebraic methods, and it is a fundamental approach to solving similar problems in algebra and calculus.