Problem 2 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ (2 x+1)(x+1)=2 x^{2} \quad ?+1 $$

Step-by-Step Solution

Answer

The completed equation is \(2x^2 + 3x + 1 = 2x^2 + 3x + 1\).

Step 1: Expanding the left part of the equation

Expand the left side of the equation using the distributive law of multiplication over addition: \( (2x+1)(x+1) = 2x^2 + 2x + x + 1 = 2x^2 + 3x + 1 \).

Step 2: Comparing the matched equation with the given equation

Compare the resulting expression \(2x^2 + 3x + 1\) with the given equation \(2x^2 + ? + 1\). It's apparent that the term \(3x\) corresponds to the missing term denoted by '?' in the provided equation.

Step 3: Writing the completed equation

Replace '?' with the found term \(3x\) in the original equation, yielding the completed equation: \(2x^2 + 3x + 1 = 2x^2 + 3x + 1\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used when multiplying a single term by two or more terms inside a set of parentheses. It states that multiplying a number by a sum is the same as multiplying the number by each addend individually and then adding the results. For example, if you have

• \(a(b + c) = ab + ac\)

In the context of polynomials, the distributive property helps simplify expressions and solve more complex equations. It's particularly useful in expanding expressions, where each term inside one set of parentheses is multiplied by every term inside another set. In the original exercise, the expression

• \((2x + 1)(x + 1)\)

requires the distributive property to combine all terms step-by-step, starting with

• \(2x\times(x + 1) + 1\times(x + 1)\).

Expanding Expressions

Expanding expressions involves using the distributive property to remove parentheses and write out a polynomial in its complete form. This step is crucial when working with algebraic equations, as it breaks expressions down into simpler components. To expand an expression like

• \((2x + 1)(x + 1)\),

you apply the distributive property by individually multiplying each term from one set of parentheses with each term from the other. This process looks like:

• \(2x \cdot x = 2x^2\)
• \(2x \cdot 1 = 2x\)
• \(1 \cdot x = x\)
• \(1 \cdot 1 = 1\)

Adding these products together provides the expanded form:

• \(2x^2 + 2x + x + 1\)

This step is essential in solving polynomials and further simplifies equations, allowing you to identify and arrange terms as needed.

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically represented in the form

• \(ax^2 + bx + c = 0\),

where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. The highest power of \(x\) is squared, which characterizes the equation as quadratic. Quadratic equations often arise in algebra problems involving area, physics, and finance, among other fields.
In the solution process, once we used the distributive property and expanded the expression, we identified terms that form the quadratic equation. In our example,

• \(2x^2 + 3x + 1 = 2x^2 + ? + 1\),

the resulting expanded equation aligns with the standard form of a quadratic equation. Solving quadratic equations can involve various methods, such as factoring, using the quadratic formula, or completing the square, which helps find the values of \(x\) that satisfy the equation.