Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Show that the equation is an identity. $$(3 x-4)(2 x+1)+5 x=6 x^{2}-4$$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to the same expression, \(6x^2 - 4\).

Step 1: Expand the Left Side

First, let's expand the product on the left-hand side of the equation:\((3x-4)(2x+1) = (3x \cdot 2x) + (3x \cdot 1) - (4 \cdot 2x) - (4 \cdot 1)\)Simplify each term:\( = 6x^2 + 3x - 8x - 4\)Combine like terms:\(= 6x^2 - 5x - 4\)

Step 2: Add 5x to the Expanded Expression

Now, add the term \(5x\) to the expanded expression from Step 1:\(6x^2 - 5x - 4 + 5x\)Combine the like terms \(-5x\) and \(+5x\):\(= 6x^2 - 4\)

Step 3: Compare Both Sides of the Equation

Now, compare the simplified left side to the right side of the original equation:\(6x^2 - 4\) (left side after simplifying) equals \(6x^2 - 4\) (right side given in the problem).Since both sides are the same, the equation is indeed an identity.

Key Concepts

Expanding Expressions

Expanding expressions is a fundamental concept in algebra that involves multiplying out brackets to simplify an expression. In our original exercise, we dealt with the expression \((3x-4)(2x+1)\). To expand this, we used the distributive property, which means each term in the first bracket must be multiplied by each term in the second bracket.Here's how it works step by step:

• First, multiply the first term of the first bracket by each term in the second bracket: \(3x \cdot 2x\) and \(3x \cdot 1\).
• Then, multiply the second term of the first bracket by each term in the second bracket: \(-4 \cdot 2x\) and \(-4 \cdot 1\).
• Combine all these products together: \(6x^2 + 3x - 8x - 4\).

Remember to handle positive and negative signs correctly, as they can change the term's contribution in the expression. By following these steps, you ensure that you've fully expanded the expression!

Combining Like Terms

Once an expression is expanded, it's important to simplify it by combining like terms. Like terms have the same variables raised to the same power. In the expanded expression \(6x^2 + 3x - 8x - 4\), the terms \(3x\) and \(-8x\) are like terms, because they both contain an \(x\) to the first power.To combine them, simply perform the operation indicated by their coefficients. So we'll perform: \(3x - 8x\), which equals \(-5x\).Our resulting expression becomes:

• \(6x^2 - 5x - 4\)

This step streamlines the expression, making it easier to manage and simplifies equivalence checks when solving equations!

Equations

Equations are fundamental in algebra, offering a statement of equality between two expressions. They are solved by manipulating one or both sides to maintain this equality. In the original solution, our equation involved simplifying the left-hand side \((3x-4)(2x+1) + 5x\) and comparing it to the right-hand side \(6x^2 - 4\).Here's how we proceed:

• First, expand the products on the left-hand side to get \(6x^2 - 5x - 4\).
• Next, add the \(5x\) term included in the original problem:
• Combine \(-5x\) and \(5x\) to simplify further, arriving at \(6x^2 - 4\).

This sequence of operations leads us to see that after simplification, both sides of the equation become equal, reinforcing the initial equality proposed by the equation.

Proofs of Identities

Proving an algebraic expression is an identity requires showing that both sides of an equation are always equal for any value that the variables can take. In our case, the expression \((3x-4)(2x+1) + 5x = 6x^2 - 4\) needed verification.An identity holds because, after expansion and simplification, each side of the equation yields the same result. For our expression:

• The left-hand side reduces to \(6x^2 - 4\), after expanding and simplifying, using like terms.
• The right-hand side was already \(6x^2 - 4\).

By demonstrating that both sides are equal, we can confidently claim that the equation is indeed an identity. This is a critical process in algebra as it helps confirm the consistency and correctness of mathematical expressions in broader applications.