Problem 27 - Algebra 2 and Trigonometry

Question

Solve for the variable and check. Each solution is an integer. \((x+3)^{2}=(x-5)^{2}\)

Step-by-Step Solution

Answer

The solution is \(x = 1\), and it satisfies the equation when checked.

Step 1: Understand the Equation

We are given the equation \((x+3)^2 = (x-5)^2\). This equation means that the square of \(x+3\) is equal to the square of \(x-5\).

Step 2: Apply Square Roots

Since both sides are squares, we can apply the square root to both sides to drop the squares: \(x+3 = x-5\) or \(x+3 = -(x-5)\).

Step 3: Solve the First Equation

From \(x+3 = x-5\), this simplifies to \(3 = -5\), which is inconsistent and provides no solution.

Step 4: Solve the Second Equation

From \(x+3 = -(x-5)\), simplify this to get \(x+3 = -x+5\). Adding x to both sides gives \(2x = 5 - 3\), which is \(2x = 2\). Thus, divide by 2 to get \(x = 1\).

Step 5: Verify the Solution

Substitute \(x=1\) back into the original equation: \((1+3)^2 = (1-5)^2\). This results in \(4^2 = (-4)^2\), which simplifies to \(16 = 16\). The solution checks out.

Key Concepts

Applying Square Roots

When solving an equation like \((x+3)^2 = (x-5)^2\), we can streamline the process by applying square roots. This allows us to "remove" the squares from both sides, which simplifies the equation significantly.

• By taking the square root of both sides, we extract the expressions inside the squares: \((x+3) = (x-5)\) or \((x+3) = -(x-5)\).
• The reason we get two options is due to the property of squaring: both positive and negative values have the same squares (e.g., \(3^2 = (-3)^2\)).

This dual nature is crucial for solving these equations. Remember to check both options, as it may lead to different solutions.

Checking Solutions

Once a solution is found, it's important to verify it by substituting the result back into the original equation. This ensures that the solution satisfies the initial conditions.
For example, consider that we found \(x=1\) in the equation \((x+3)^2 = (x-5)^2\).

• We substitute \(x=1\) back into the equation, resulting in \( (1+3)^2 = (1-5)^2 \).
• This simplifies to \(4^2 = (-4)^2\), or \16 = 16\.

Since both sides are equal, \(x=1\) is confirmed as the correct solution.
Checking our work is an essential habit in math, helping catch any errors and reinforcing the validity of the solution.

Integer Solutions

Integer solutions are whole numbers that satisfy an equation. In the given exercise, the solution \(x=1\) is an integer, which aligns with the problem's requirement that each solution must be an integer.
Why are integers significant here? They keep calculations simple and often make sense in practical scenarios, like counting items or steps.

• When solving equations, you may end up with fractions or decimals. If a problem specifies that solutions must be integers, it's vital to ensure the solutions adhere to this format.

For the exercise, the consistency between the problem statement and our integer solution demonstrates correct alignment with the task requirements.

Simplification of Equations

Equation simplification is the process of transforming expressions into their simplest form. This makes equations easier to handle and solves them effectively. In our exercise:

• We start with the complex equation \( (x+3)^2 = (x-5)^2 \).
• Applying square roots simplifies it to two possible equations: \(x+3 = x-5\) and \(x+3 = -(x-5)\).
• The first equation \(3 = -5\) is impossible and provides no integers, which simplifies our workload by eliminating it.
• The second equation simplifies to \(2x = 2\), leading directly to \(x = 1\).

Each step reduces complexity, making it easier to find correct solutions efficiently.
Simplifying before solving can reveal insights about the character of the solutions and streamline problem-solving.