Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ (2 x+1)^{2}=4\left(x^{2}+x+1\right) $$

Step-by-Step Solution

Answer

The given quadratic equation has no solutions as it simplifies down to 1 = 4, which is a contradiction.

Step 1: Simplify and rearrange the equation

Start by expanding the square of the binomial and the right side of equation. The equation becomes: \(4x^2 + 4x + 1 = 4x^2 + 4x + 4\). Then, subtract \(4x^2 + 4x\) from both sides to consolidate like terms. This step results in \(4x^2 + 4x + 1 - 4x^2 - 4x = 4x^2 + 4x + 4 - 4x^2 - 4x\). So, \(1 = 4\)

Step 2: Check for solutions

Since both sides are constants and the left side is not equal to the right side, this equation has no solution.

Step 3: Verification

There's no need to verify as there is no solution to this equation, as found in step 2.

Key Concepts

Equation Solving

Solving equations is like solving a mystery where you want to find the value of the unknown, often represented by a variable like \(x\). The process involves simplifying the equation until you can isolate the variable on one side of the equation. In our exercise, we're dealing with a quadratic equation. A quadratic equation is a polynomial equation of degree 2, which means the highest power of the variable is squared (\(x^2\)). Here, the equation starts with expanding both sides to make it more manageable. We expand \((2x+1)^2\) and the right-hand side to get two equivalent expressions. Then, we simplify by collecting like terms and subtract from both sides if necessary. Unlike many puzzles that have specific answers, some equations might not have a solution at all, which leads us to our next concept.

No Solution

An equation with 'no solution' can be puzzling because it suggests there's no value for the variable that makes the equation true. This happens when, after simplifying the equation, the remaining statement is a contradiction, such as \(1 = 4\). This means that there are no values of \(x\) that will satisfy the equation.

In the exercise, after simplifying everything, we found \(1 = 4\), which is impossible. The equation becomes an identity that is false, indicating it has no solution. Not all mathematical problems have solutions, especially when inconsistencies like these arise. All mathematical steps might structurally seem correct but can still lead to a contradiction in some cases.

Algebraic Techniques

When tackling quadratic equations, various techniques help in simplifying and solving them. One common method is expanding binomials, which involves multiplying out expressions like \((2x+1)^2\). This expansion uses the formula

• \((a+b)^2 = a^2 + 2ab + b^2\)

Understanding this technique is critical in simplifying equations.

Once expanded and when the equation is balanced, the next step is to combine like terms to simplify it further. Combining like terms involves adding or subtracting coefficients of the same power of the variable together. If after doing so, both sides still don’t balance out, then you might have an equation with no solution.

Algebraic techniques serve as powerful tools for equation solving, revealing hidden intricacies and sometimes, even showing that a solution does not exist.