Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ (2 x-1)^{2}=4\left(x^{2}-x+6\right) $$

Step-by-Step Solution

Answer

The given equation has no solution because the simplification of the equation gives a wrong equality i.e. \(0 = 23\) which is not possible.

Step 1: Simplify the Equation

Simplify the equation by expanding the squares on both the sides. \[(2x-1)^2 = 4x^2 - 4x + 24\]Expanding gives:\[4x^2 - 4x + 1 = 4x^2 - 4x + 24\]

Step 2: Simplify Further

Further simplify the equation by subtracting \(4x^2\) and adding \(4x\) to both sides to eliminate \(x\)'s from one side, giving:\[0 = 23\]

Step 3: Check for Solution

On simplifying, we see that equation gives a result that \(0 = 23\), which is incorrect. Therefore, the initial equation given has no solution.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are the bread and butter of high school mathematics, providing a critical foundation for understanding complex mathematical concepts. At its core, an algebraic equation is a mathematical statement that asserts the equality of two expressions with variables, numbers, and operations such as addition and multiplication. These variables, usually denoted by letters like x or y, act as placeholders for numbers we want to find, known as unknowns.

When solving algebraic equations, the goal is to isolate the variable on one side of the equation in order to find its value. This process typically involves a series of steps that can include combining like terms, using the distributive property, and performing operations that balance both sides of the equation. The complexity of these steps increases with the equation's complexity, as seen in our exercise where a squared binomial expands to reveal an equation with no solution.

The Art of Equation Simplification

Equation simplification is a process that reduces equations to more manageable forms, making them easier to solve. This can involve expanding expressions, canceling out terms, and combining like terms. Simplification often includes the use of distributive properties to expand products and eliminate parentheses, as we've observed in the exercise where \(2x - 1)^2\) was expanded to \(4x^2 - 4x + 1\).

In our example, further simplification was attempted by eliminating the \(4x^2\) and \(4x\) terms on both sides, aiming to isolate the variable. However, this led to an impossible statement \(0 = 23\), signifying that the equation has no solution. This illustrates an important aspect of simplification: it not only aids in solving equations but also in recognizing when equations are unsolvable.

Quadratic Formulas and Their Uses

When algebra takes us beyond linear equations, we are often faced with quadratic equations, which are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These are second-degree equations because the variable is raised to the power of two, hence the term 'quadratic'.

To solve these types of equations, we typically use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula provides the solutions for \(x\) by plugging in the values of \(a\), \(b\), and \(c\) from the quadratic equation. In certain cases, like the exercise we're discussing, equations may resemble a quadratic format but not yield any real solutions, as the simplification leads to a nonsense equation. This is an important concept to recognize, as not all quadratic-like equations have solutions that are real numbers.