Problem 108 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$(x-5)^{2}=x^{2}-5 x+25$$

Step-by-Step Solution

Answer

The statement is False as written. The correct statement should be \((x-5)^{2}=x^{2}-10x+25\).

Step 1: Expansion of Left Side

First, we expand \((x-5)^{2}\) to get \(x^{2}-2*5*x+25 = x^{2}-10x+25\)

Step 2: Comparison with Right Side

We then compare the expanded equation with \(x^{2}-5 x+25\). The difference is in the second term where the left side has \(10x\) and the right side has \(5x\).

Step 3: Modification of Statement

To make the statement true we have to change the coefficient of \(x\) in the right side from \(5x\) to \(10x\). Hence the correct equation should be \(x^{2}-10x+25\).

Key Concepts

Expanding Algebraic Expressions

In algebra, expanding expressions involves multiplying out the terms and simplifying them into a longer expression. This is particularly important when dealing with expressions inside parentheses raised to a power. For instance, consider

• \((x-y)^2\)

To expand, you apply the distributive property:

• Calculate \((x-y) \times (x-y)\)
• Distribute each term: \((x \times x) - (2xy) + (y \times y)\)
• Simplify: \(x^2 - 2xy + y^2\)

This process unpacks the expression into a more workable form. You'll need this skill to simplify and solve complex algebraic equations.

Comparing Algebraic Expressions

Comparing expressions means seeing if two expressions are identical or equivalent. You do this by simplifying both expressions to see if they match. For example in the step-by-step solution, we expanded:

• Left: \(x^2 - 10x + 25\)
• Right: \(x^2 - 5x + 25\)

The difference is in the coefficients of \(x\), leading to a difference in values. Knowing how to identify these differences helps you decide the necessary adjustments to make the expressions match.

Correcting Errors in Equations

Correcting errors in equations involves identifying the discrepancies and making the necessary changes to correct them. In the provided problem, we saw that:

• The expanded term was \(-10x\), whereas the comparison term had \(-5x\).
• To correct this, adjust the term from \(-5x\) to \(-10x\) on the mistaken side.

This type of correction is crucial for ensuring that both sides of an equation or comparison align perfectly, establishing the true equivalence between the algebraic expressions.