Problem 79 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$2 x^{2}+4\left(3 x-x^{2}\right)+3 x$$

Step-by-Step Solution

Answer

The expression simplifies to \(-2x^2 + 15x\).

Step 1: Distribute the 4

First, distribute the 4 into the terms inside the parentheses: \[ 4(3x - x^2) = 4 imes 3x - 4 imes x^2 = 12x - 4x^2 \]This simplifies the expression to:\[ 2x^2 + 12x - 4x^2 + 3x \]

Step 2: Combine Like Terms

Now, combine the like terms by grouping the expressions with the same variable part:1. Combine the \( x^2 \) terms: \[ 2x^2 - 4x^2 = -2x^2 \]2. Combine the x terms: \[ 12x + 3x = 15x \]This results in the simplified expression:\[ -2x^2 + 15x \]

Key Concepts

Distributive Property

The distributive property is a key mathematical principle that makes simplifying expressions much easier. Put simply, it tells us how to multiply a single term by each term within a set of parentheses. For example, in our exercise, we need to distribute the 4 across both terms inside the parentheses \((3x - x^2)\).

Here's how it works:- Multiply 4 by \(3x\), which gives you \(12x\)- Multiply 4 by \(-x^2\), resulting in \(-4x^2\)After distributing, you can take a deep breath because now our original expression \(2x^2 + 4(3x - x^2) + 3x\) finds itself simplified to \(2x^2 + 12x - 4x^2 + 3x\).

This clever property helps break down and untangle complex-looking problems into simpler forms that we can more easily manage.

Combining Like Terms

Combining like terms is a fundamental step in simplifying a polynomial. This means putting together terms that have the same variable raised to the same power. Finding these groups can help make the expression a lot more compact.

In our exercise, there are two groups of like terms:

• The \(x^2\) terms: \(2x^2\) and \(-4x^2\)
• The x terms: \(12x\) and \(3x\)

Let's tackle each group:- For the \(x^2\) terms, \(2x^2 - 4x^2\) simplifies to \(-2x^2\).- For the x terms, \(12x + 3x\) combines to give us \(15x\).Thus, like a puzzle coming together, we now have a simplified expression: \(-2x^2 + 15x\).

This is how combining like terms works its magic: it streamlines and declutters a polynomial.

Quadratic Expression

A quadratic expression is a polynomial that involves the square of a variable, usually represented in the form \(ax^2 + bx + c\). Our final simplified expression from the exercise, \(-2x^2 + 15x\), is a perfect example of a quadratic expression.

Here's why this is important:- The term \(-2x^2\) is the quadratic term since it features \(x^2\).- The term \(15x\) is the linear term because it involves just \(x\).- These types of expressions are pivotal because they show up in various real-world phenomena, like calculating areas, modeling paths of objects, and finding maximum and minimum values.

Quadratic expressions can be more than just numbers and letters; they form equations and functions that describe real, everyday situations.