Problem 136 - Algebra and Trigonometry

Question

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. $$x^{2}+36=(x+6)^{2}$$

Step-by-Step Solution

Answer

The statement \(x^{2}+36=(x+6)^{2}\) is false. The correct statement is \(x^{2}+12x+36=(x+6)^{2}\).

Step 1: Expanding the Right Side of the Equation

Expand the right side of the equation which is \((x+6)^{2}\), using the formula \(a^2+2ab+b^2\). This will give \(x^{2}+12x+36\). The equation now becomes \(x^{2}+36=x^{2}+12x+36\).

Step 2: Simplifying the Expression

Now, simplify the equation by subtracting \(x^{2} + 36\) from both sides. This will give '0' on the left side and 12x on the right side. This equation now takes the form of \(0=12x\). This implies x = 0.

Step 3: Evaluating the Truth Value of the Original Statement

The original statement, \(x^{2}+36=(x+6)^{2}\), is only true for \(x=0\). For other values of \(x\), the equation does not hold, hence the original statement is false.

Step 4: Correcting the Original Statement

To correct the original statement, a term '12x' would need to be added to the left side of the equation. The corrected equation becomes \(x^{2}+12x+36=(x+6)^{2}\). Now this equation holds true for all values of \(x\).

Key Concepts

Polynomial Expansion

In algebra, polynomial expansion is key to breaking down complex expressions. When we expand \((x+6)^{2}\), we're using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a\) is \(x\), and \(b\) is \(6\).

• First, calculate \(x^2\), which stays as it is.
• Next, multiply \(2 \times x \times 6\) to get \(12x\).
• Lastly, find \(6^2\), which equals 36.

Putting it all together, the expanded form of \((x+6)^2\) is \(x^2 + 12x + 36\). Expanding polynomials this way makes it easier to compare equations.

Quadratic Equations

A quadratic equation is any equation that can be rearranged in the form \(ax^2 + bx + c = 0\). Our problem involves understanding if \(x^2 + 36\) is equal to an expanded quadratic form.

• The left side, \(x^2 + 36\), lacks a linear term, \(bx\).
• Meanwhile, \((x+6)^2\) expands to \(x^2 + 12x + 36\), indicating a full quadratic expression.

By comparing both sides, it's clear that the left side is missing the \(12x\) term, showing how equations can be incomplete or assumed wrongly.

Equation Simplification

Simplifying equations allows us to find values of unknowns easily. Taking the given equation \(x^2 + 36 = x^2 + 12x + 36\):

• Subtract \(x^2 + 36\) from both sides to isolate terms.
• This results in \(0 = 12x\).
• From this, we deduce \(x = 0\).

Through simplification, we've isolated \(12x\), showing how reducing equations helps identify specific values of variables.

Truth Value Evaluation

Evaluating the truth of algebraic statements ensures accuracy. Initially, the statement \(x^2 + 36 = (x+6)^2\) seems plausible:

• By simplification, we found it true only when \(x = 0\).
• For other values, the equation is false because of the missing \(12x\) term.

To correct it, add \(12x\) to both sides, rewriting it as \(x^2 + 12x + 36 = (x+6)^2\). Now, it's valid for any \(x\), showcasing how truth evaluation leads to proper equation balancing.