Problem 79 - Algebra A Combined Function

Question

Fill in the squares so that each is a true statement. $$ 3 x^{\square}+4 x^{2}=7 x^{\square} $$

Step-by-Step Solution

Answer

The exponent squares should be filled with 0.

Step 1: Analyze the Given Expression

The given expression is \(3x^{\square} + 4x^2 = 7x^{\square}\). This expression involves two terms with unknown exponents shown as \(\square\). We need to fill in these squares correctly so that the equation is balanced.

Step 2: Distribute Terms for Comparison

To solve for the unknown exponents, note that the base of each term, \(x\), must have exponents that allow the equation to equalize. We want each term on the left side of the equation to match with a term on the right.

Step 3: Equate Exponents

For both sides of the equation to be identical, \(x^{\square}\) must equal the same expression on both sides. Therefore, we equate the exponents of both sides: \(3x^{\square} = 7x^{\square}\). This implies that both terms have an exponent of \(\square = 0\).

Step 4: Solve for the Square

If \(\square = 0\), then \(3x^0 + 4x^2 = 7x^0\) becomes 3 + \(4x^2 = 7\). This forms a solvable equation, reinforcing that \(\square = 0\) is correct, as it matches the structure of the equation consistently.

Key Concepts

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number by itself. For example, when you see something like \(x^2\), it indicates \(x\) multiplied by itself, or \(x \times x\). This is one of the key tools in algebra that helps to simplify and solve equations with higher powers.
When solving problems with exponents, remember:

• The base is the number that is being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• Any number raised to the power of 0 is 1 (e.g., \(x^0 = 1\)).

In the exercise, recognizing that when \(\square = 0\), \(x^0 = 1\), allows us to simplify terms and focus on simplifying the equation further. Understanding these basic properties of exponents is crucial for balancing equations, as you'll be seeing in the next sections.

Balancing Equations

Balancing equations is like keeping a scale even, where each side of the equation is perfectly matched in value. This is vital in algebra to ensure that what is manipulated or simplified keeps the expression valid and true. In practice:

• Whatever operation you do to one side, you must do to the other side.
• In the example where \(3x^\square + 4x^2 = 7x^\square\), both sides need to have the same total value after simplification.
• Balancing involves matching terms with common bases and equalizing their exponents when possible.

By filling the squares correctly, you ensure that each calculation and simplification maintains the integrity of the equation, leading to a correct solution.

Distributive Property

The distributive property is a crucial algebraic property that connects addition and multiplication. It allows you to simplify and solve equations by expanding terms and grouping them for easier manipulation. The general form is \(a(b + c) = ab + ac\).
Here's how it works practically:

• "Distribute" means to multiply the term outside the parenthesis by each term inside.
• It is frequently used to simplify expressions and make solving equations more straightforward.

In our example, understanding the distributive property can help recognize opportunities to simplify portions of equations that involve multiplication sums. While it might not be needed to directly solve the given equation, it's foundational for rearranging terms and ensuring all simplifications adhere to the properties of numbers and operations.