Problem 61 - Algebra and Trigonometry

Question

An Algebra Error Beginning algebra students sometimes make the following error when squaring a binomial: $$ (x+5)^{2}=x^{2}+25 $$ (a) Substitute a value for \(x\) to verify that this is an error. (b) What is the correct expansion for \((x+5)^{2} ?\)

Step-by-Step Solution

Answer

(a) It's an error because 49 ≠ 29; (b) Correct: \(x^2 + 10x + 25\).

Step 1: Substitute a value for x

Let's use \( x = 2 \) as our test value. Substitute \( x = 2 \) into the expression \((x + 5)^2 = x^2 + 25\) to verify the error.Calculate both sides: Left side: \((2 + 5)^2 = (7)^2 = 49\)Right side:\(2^2 + 25 = 4 + 25 = 29\)Since \(49 eq 29\), this confirms that the initial equation \((x + 5)^2 = x^2 + 25\) is incorrect.

Step 2: Recall the Binomial Theorem

To correctly expand \((x + 5)^2\), recall the binomial theorem for squaring a binomial:\((a + b)^2 = a^2 + 2ab + b^2\). In the given problem, \(a = x\) and \(b = 5\).

Step 3: Apply the Binomial Expansion

Substitute \(a = x\) and \(b = 5\) into the binomial expansion formula:\((x + 5)^2 = x^2 + 2(x)(5) + 5^2\).

Step 4: Simplify the Expression

Simplify the expression:\(x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25\).This is the correct expanded form of \((x + 5)^2\).

Key Concepts

Algebra Errors

Committing algebra errors can be quite common, especially when you are new to the subject. One frequent mistake involves squaring binomials incorrectly. For instance, some may assume that \((x+5)^2\) simplifies directly to \(x^2 + 25\). This is not correct and stems from a misunderstanding of the binomial expansion rules. To avoid these errors, always double-check your work, and remember that squaring a binomial involves more than just squaring each individual term. Use the rules for multiplication and distribution to guide your process.

Algebraic Substitution

Algebraic substitution is a valuable technique used to check the validity of algebraic expressions. To illustrate, consider substituting a value into an equation to determine whether an algebra error exists. In the given problem, by substituting \(x = 2\) into \((x+5)^2 = x^2 + 25\), we compute:

• Left side: \((2+5)^2 = 49\)
• Right side: \(2^2 + 25 = 29\)

Since these two results do not match, the original equation is incorrect. Substitution allows you to verify or disprove equations by providing tangible numerical results that confirm your understanding.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that simplifies the process of expanding expressions raised to a power. When squaring a binomial like \((x+5)^2\), the theorem creates a blueprint for the expansion without directly multiplying everything out:\[(a + b)^2 = a^2 + 2ab + b^2\]This general formula tells you to square the first term \(a\), multiply the two terms \(a\) and \(b\) together and double the product, and finally square the last term \(b\). Applying the Binomial Theorem reduces algebra errors and allows for quick, efficient expansion of expressions.

Squaring Binomials

Squaring binomials involves applying the Binomial Theorem to finetune your expressions further. When you have an expression like \((x+5)^2\), you can expand it using:

• First, square the first term: \(x^2\)
• Second, double the product of both terms: \(2 \cdot x \cdot 5 = 10x\)
• Finally, square the last term: \(5^2 = 25\)

Combine these parts to get the complete expanded form: \(x^2 + 10x + 25\). This comprehensive approach ensures that no terms are left out or incorrectly calculated, providing accurate results every time.