Problem 60 - A Graphical Approach to College Algebra

Question

Incorporate many concepts from Chapter 3 with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Let \(y_{1}=\sqrt[3]{4 x-4}\) and let \(y_{2}=\sqrt{x+1} .\) Graph \(y_{3}=y_{1}-y_{2}\) in the window \([-2,20]\) by \([-0.5,0.5]\) to determine the number of real solutions of the original equation.

Step-by-Step Solution

Answer

The graph shows one real solution.

Step 1: Substitute and Identify

We start by substituting \( y_1 = \sqrt[3]{4x - 4} \) and \( y_2 = \sqrt{x + 1} \) in the given equation. This yields the expression for \( y_3 = y_1 - y_2 = \sqrt[3]{4x - 4} - \sqrt{x + 1} \). We are tasked with graphing \( y_3 \) to identify the number of real solutions.

Step 2: Graphing \( y_3 = y_1 - y_2 \)

To determine the number of real solutions, graph \( y_3 = \sqrt[3]{4x - 4} - \sqrt{x + 1} \) over the interval \([-2, 20]\) for the x-axis and \([-0.5, 0.5]\) for the y-axis. The solutions to the equation correspond to points where the graph of \( y_3 \) crosses the x-axis.

Step 3: Analyze the Graph

Upon graphing, observe the points where the curve \( y_3 \) intersects the x-axis (i.e., the line \( y_3 = 0 \)). Each intersection denotes a real solution to the original equation, as it indicates where \( \sqrt[3]{4x - 4} = \sqrt{x + 1} \).

Key Concepts

Roots of Equations

In mathematics, the concept of finding the 'roots' of an equation is crucial. A root of an equation is essentially the value of the variable that makes the equation true. Imagine you have an equation where an expression with a variable is equal to zero. The solution, or the root, is the specific value which satisfies this.
One way of finding roots is through substitution techniques. For instance, in the original exercise, we substitute parts of the given equation with simpler terms - namely, \( y_1 \) and \( y_2 \). This allows us to focus on the equation \( y_3 \) derived from \( y_1 - y_2 \). By simplifying the expressions, we can identify where this expression equates to zero, giving us the roots of the original equation.
The roots are crucial as they tell us about the behavior of the equation over the set of real numbers. Essentially, by finding these roots, we can determine at which points the expressions on either side of the equation are equal.

Graphical Analysis

Graphical analysis offers a visual method for determining the solutions to an equation. This is particularly useful with equations involving complex expressions, like those with roots. When we graph an equation like \( y_3 = y_1 - y_2 \), we seek to identify where this graph intersects the x-axis.
This intersection is significant because it visually represents the roots of the equation. When the graph of a function crosses the x-axis, it means the output (or 'y' value) at that point is zero. For \( y_3 \), this indicates that \( \sqrt[3]{4x - 4} \) and \( \sqrt{x + 1} \) are equal.
Through plotting the graph over a specified domain and range — like the interval \([-2, 20]\) for x, and \([-0.5, 0.5]\) for y in the exercise — we can clearly see these intersections. Such graphical solutions make it easier for students to understand equations visually and conceptually.

Real Solutions

Real solutions refer to those solutions of an equation that are real numbers, as opposed to complex or imaginary numbers. In the context of our exercise, identifying real solutions involves finding where the expressions on both sides of the equation truly match, resulting in a valid equality on the real number line.
In practical terms, a real solution is where two graphed expressions intersect along the x-axis on a 2D plane. By setting \( y_3 = 0 \) and plotting its function, we determine when the cube root and square root expressions in the original problem are equal.
This means checking when \( \sqrt[3]{4x - 4} \) equals \( \sqrt{x + 1} \) within the real number system, creating an opportunity to physically see how many times and at which values this occurs. Real solutions ensure that the roots we find are applicable in everyday scenarios and calculations, which is often vital for practical mathematical applications.