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Chapter 1: Problem 92

Solve the equation by using any method. . \(p^{2}-4 p=4 p(p-1)-3 p^{2}+2\)

Short Answer

Expert verified
No solution, the equation is a contradiction.

Step by step solution

01

Expand and Simplify the Right-Hand Side

First, expand the right-hand side of the equation: \[ p^2 - 4p = 4p(p - 1) - 3p^2 + 2 \].Distribute the terms to get: \[ p^2 - 4p = 4p^2 - 4p - 3p^2 + 2 \].
02

Combine Like Terms

Next, combine like terms on the right-hand side: \[ p^2 - 4p = (4p^2 - 3p^2) - 4p + 2 \].This simplifies to: \[ p^2 - 4p = p^2 - 4p + 2 \].
03

Move All Terms to One Side

Subtract \(p^2 - 4p\) from both sides of the equation to get: \[ p^2 - 4p - (p^2 - 4p) = p^2 - 4p + 2 - (p^2 - 4p) \].This simplifies to: 0 = 2.
04

Analyze the Simplified Equation

Since the simplified equation 0 = 2 is a contradiction, it implies that there is no value of \( p \) that satisfies the original equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expand and simplify
When solving quadratic equations, expanding and simplifying is an important step. It means distributing and breaking down the terms so that you can combine like terms easily. In the given exercise, the equation is: \[ p^2 - 4p = 4p(p - 1) - 3p^2 + 2 \]To expand, distribute the terms on the right-hand side: \[ p^2 - 4p = 4p^2 - 4p - 3p^2 + 2 \]Now, the terms are expanded, and we are ready to move on to combining like terms. This will help us further simplify the equation and understand it better.
Combining like terms
Combining like terms means adding or subtracting terms with the same variable and the same exponent. From the expanded equation: \[ p^2 - 4p = 4p^2 - 4p - 3p^2 + 2 \]Identify and combine the like terms: \[ p^2 - 4p = (4p^2 - 3p^2) - 4p + 2 \]This simplifies to: \[ p^2 - 4p = p^2 - 4p + 2 \]By combining like terms, we make solving equations cleaner and more straightforward. It reduces the equation to simpler and more manageable components.
Contradiction in equations
Sometimes, when solving equations, you might end up with a statement that doesn't make sense鈥攖his is called a contradiction. In our simplified equation: \[ p^2 - 4p = p^2 - 4p + 2 \]Subtracting \( p^2 - 4p \) from both sides yields: \[ 0 = 2 \]Since 0 doesn't equal 2, this equation shows a contradiction. A contradiction implies that there are no values of the variable that can satisfy the equation. In other words, the original equation has no solution.
Zero-product property
The zero-product property is a crucial concept for solving quadratic and polynomial equations. It states that if the product of two numbers is zero, at least one of the numbers must be zero. Mathematically: \[ ab = 0 \implies a = 0 \text{ or } b = 0 \]Using the zero-product property allows us to break down complex equations into simpler ones. Although the exercise equation didn't directly utilize this property, understanding it is fundamental for other quadratic equations where you factorize to find the roots. This concept helps in solving and breaking down the equation into manageable parts where each part equals zero and can be solved separately.

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