Solving a Complex Equation: A Comprehensive Guide

Introduction

In this article, we will guide you through solving a complex mathematical equation step by step. The equation in question is:

16 times 32x 2 5 times 2x

We will break down the problem into smaller, manageable parts, using algebraic substitution and calculus.

Solving the Equation

The equation can be written as:

16 times (2x)5 2 5 times 2x

Let's make the substitution:

y 2x

Substituting y into the equation, we get:

16y5 2 5y

Which simplifies to:

16y5 5y 2 0

This is a fifth-degree polynomial in y. To solve for y, we need to find the roots of this polynomial. We will use the Rational Root Theorem to identify potential rational roots.

Applying the Rational Root Theorem

The Rational Root Theorem states that any rational root, expressed as a fraction p/q, is such that p is a factor of the constant term (here, -2) and q is a factor of the leading coefficient (here, 16).

The factors of -2 are ±1, ±2. The factors of 16 are ±1, ±2, ±4, ±8, ±16. Therefore, the possible rational roots are ±1, ±1/2, ±1/4, ±1/8, ±1/16, ±2, ±2/16 ±1/8.

Let's test some of these possible roots:

y 1/2

Substituting y 1/2 into the polynomial:

16(1/2)5 5(1/2) 2 0

This simplifies to:

16(1/32) 5/2 2 0

1/2 5/2 2 0

1 5 4 0

0 0

This confirms that y 1/2 is a root.

Graphical Analysis

To better understand the polynomial, we can make a rough sketch or plot of the function z 16y5 5y 2. We need to understand the function's behavior with a little bit of calculus.

Step 1: Finding Turning Points

To find the turning points, we take the derivative of the polynomial:

z' 80y4 5

Setting the derivative equal to zero:

80y4 5 0

y4 1/16

y ±1/2

Step 2: Finding the z-coordinates of the Turning Points

Substitute y 1/2 into the polynomial:

z 16(1/2)5 5(1/2) 2 0

z 1/2 5/2 2 0

Z 0

Substitute y -1/2 into the polynomial:

z 16(-1/2)5 5(-1/2) 2

Z -1/2 5/2 2 4

Step 3: Summary of the Function

Given the leading coefficient and the degree of the polynomial as odd, the function has a turning point at -1/2, 4 and 1/2, 0. The function increases from the bottom, reaches the turning point, and then decreases, before increasing again.

The turning points help us understand the behavior of the polynomial, confirming that the only real solution is:

y 1/2

Converting y back to x:

2x 1/2

x -1

Conclusion

The only real solution to the equation is x -1. This comprehensive guide illustrates the step-by-step process of solving the equation, including algebraic substitutions and graphical analysis.