Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ (x+1)^{2}+2(x-2)=(x+1)(x-2) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{1}{5}\).

Step 1: Rewrite the Equation

Rewrite each side of the equation to make it easier to simplify. This simplifies to \(x^{2} + 2x + 1 + 2x - 4 = x^{2} - x - 2\). Simplify further to obtain \(x^{2} + 4x - 3 = x^{2} - x - 2\).

Step 2: Simplify the Equation

Move all terms to one side of the equation to set it equal to zero. This provides the equation \(5x + 1 = 0\).

Step 3: Solve for x

To isolate \(x\), subtract 1 from both sides, then divide all terms by 5. This gives \(x = -\frac{1}{5}\).

Step 4: Check the Solution

Substitute \(x = -\frac{1}{5}\) back into the original equation to verify the solution. When substituted, both sides of the equation should balance, which confirms that \(x = -\frac{1}{5}\) is indeed the correct solution.

Key Concepts

Quadratic Equation Simplification

When it comes to solving quadratic equations, simplification is key to reducing complexity and making the equation more manageable. In the given exercise, simplification starts with expanding the squared terms and the products, which transforms the equation into a more straightforward format that can be easily manipulated. For example, simplifying \( (x+1)^2 \) involves squaring both the variable and the constant and then applying the distributive property to get \( x^2 + 2x + 1 \).

The next step involves expanding \( (x+1)(x-2) \) by multiplying each term in the first parentheses by each term in the second parentheses, resulting in \( x^2 - x - 2 \). This process is crucial as it turns the equation into a standard quadratic form, which allows us to combine like terms and move towards solving the equation.

• Initially, the equation may appear complicated, but expanding and simplifying can reveal a simpler form or even a linear equation.
• Ensuring that all like terms are combined helps in clearly seeing the structure of the equation and simplifies further steps.
• Always double-check your arithmetic when simplifying to prevent errors moving forward.

Simplified equations are easier to solve and often require fewer steps, making the overall process more efficient and less error-prone.

Isolate Variable Techniques

The essence of algebra is to isolate the variable of interest, and in quadratic equations, this means manipulating the equation to solve for \(x\). In our exercise, once we've simplified the equation to \(5x + 1 = 0\), we are dealing with a simpler linear equation. The 'isolate variable techniques' come into play here.

To isolate \(x\), we must first eliminate the constant term from the left side. Performing the operation \( -1 \) on both sides, we get \(5x = -1\). Next, to get \(x\) by itself, we divide both sides of the equation by the coefficient \(5\), yielding \(x = -\frac{1}{5}\).

• Remember, when manipulating equations, whatever you do to one side, you must do to the other side to maintain equality.
• Using inverse operations, such as subtracting to counter addition and dividing to counter multiplication, are fundamental in isolating variables.

This approach streamlines the process and eliminates any extraneous information, making the variable in question the sole focus.

Equation Verification Methods

After finding a potential solution to a quadratic equation, verification ensures the accuracy of the result. The 'equation verification methods' involve substituting the solution back into the original equation and confirming if the left and right sides equate. For this exercise, by plugging in \(x = -\frac{1}{5}\), we check each term carefully:

• Begin with \( (x+1)^2 \) and substitute \(x \) with \( -\frac{1}{5} \), which simplifies to a quantifiable value.
• Perform the same substitution for the right side of the equation with \( (x+1)(x-2) \) and simplify.
• After substitution, if both sides simplify to the same value, the solution is correct.

Through verification, we protect against common mistakes and reinforce our understanding of the concepts at play. It's a crucial step that not only provides a safety check but also serves as a learning tool for students to see the equation come full circle.