Solving for b in a Quadratic Equation: A Comprehensive Guide

Whether you are a student or a professional working with algebraic equations, understanding how to solve for an unknown variable such as b is a crucial skill. This guide will walk you through the process of evaluating and solving the equation x pb2/b - 1 for b. We will break down the steps required to manipulate the equation into a more solvable form and provide a thorough explanation of the algebraic manipulations involved.

Understanding the Equation

The given equation is x pb2/b - 1. Here, x and p are known constants, and b is the unknown variable we aim to solve for. Before we delve into the solution, it is imperative to establish some basic assumptions and conditions to ensure the equation is valid.

Initial Setup

To solve for b, we start with the given equation:

x  pb2/b - 1

To simplify, we note that b cannot be zero. If b 0, the equation would involve division by zero, which is undefined. Therefore, we can safely assume that b is not zero.

Manipulating the Equation

Next, we will multiply both sides of the equation by b to eliminate the denominator:

bx  pb2 - b

Now, to simplify further, we can move the term b from the right side to the left side:

bx   b  pb2

We can now factor out b from the left side:

b(x   1)  pb2

To isolate b, we divide both sides by b (again, ensuring b is not zero):

b2 - (x   1)b/p  0

Next, we recognize that this is a quadratic equation in the form of eb2 fb g 0, where e p2, f - (x 1)p, and g 0.

Solving the Quadratic Equation

The general solution to a quadratic equation eb2 fb g 0 is given by:

b1,2  -b plusmn; sqrt{f2 - 4eg}/2e

Substituting the values of e, f, and g into the equation, we get:

b1,2  (x   1)p plusmn; sqrt{((x   1)p)2 - 4p()2(0)}/2p2

Since g 0, the equation simplifies to:

b1,2  (x   1)p plusmn; sqrt{(x   1)2p2)/2p2

We can simplify the square root term:

b1,2  (x   1)p plusmn; (x   1)p/2p

This further simplifies to:

b1,2  (x   1)p plusmn; (x   1)/2

We can now have two possible solutions for b:

b1 (x 1)p (x 1)/2 (2x 2p 1)/2

b2 (x 1)p - (x 1)/2 (2x 2p - 1)/2

These are the possible values for b.

Conclusion

In conclusion, solving for b in the equation x pb2/b - 1 involves a series of algebraic manipulations including factoring, simplifying, and applying the quadratic formula. The solutions for b are given by:

b1,2  (2x   2p ± 1)/2

It is important to note the conditions under which these solutions are valid, specifically ensuring b is not zero and that the term under the square root is non-negative. By understanding these steps and ensuring all assumptions are met, you can confidently solve similar quadratic equations.

Best of luck in your mathematical journey!