Problem 49 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(3-x)^{2}=25$$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = 8\).

Step 1: Understand the Equation

The given equation is \((3-x)^2 = 25\). This is a quadratic equation written in a specific form, where we need to find the values of \(x\) that satisfy this equation.

Step 2: Take the Square Root

To solve the equation \((3-x)^2 = 25\), first take the square root of both sides. This gives us two possible solutions: one positive and one negative, since the square of a number can be either positive or negative.\[3-x = 5 \quad \text{or} \quad 3-x = -5\]

Step 3: Solve for x

Now, solve each of the two equations obtained in Step 2.For \(3-x = 5\):\[3 - x = 5 \Rightarrow -x = 5 - 3 \Rightarrow -x = 2 \Rightarrow x = -2\]For \(3-x = -5\):\[3 - x = -5 \Rightarrow -x = -5 - 3 \Rightarrow -x = -8 \Rightarrow x = 8\]

Step 4: Verify the Solutions Graphically

You can graph the equation \(y = (3-x)^2\) and \(y = 25\) to see where they intersect. The intersections represent the solutions to the equation.- The graph \((3-x)^2\) is a parabola opening upwards centered at \(x = 3\).- The line \(y = 25\) is a horizontal line.The intersections should occur at \(x = -2\) and \(x = 8\), verifying the solutions.

Key Concepts

Graphical Solutions

Solving quadratic equations graphically is a great visual tool. It helps you understand where the solutions lie on a graph. Here's how you can interpret the solution graphically for the original exercise:

• The equation \((3-x)^2 = 25\) can be split into two parts: the function \(y = (3-x)^2\) and the constant function \(y = 25\).
• To find the solutions graphically, we plot both on the same coordinate system and look for points where they intersect.
• Wherever the graph of the function \(y = (3-x)^2\) meets the line \(y = 25\), you find the solutions for \(x\).

The parabola \((3-x)^2\) represents the quadratic component and opens upwards. The line \(y = 25\) shows the horizontal line where possible solutions lie. By inspecting where the parabola intersects the line, you can visually confirm the solution from the algebraic steps (i.e., \(x = -2\) and \(x = 8\)). This verification helps in understanding not just the solution itself, but the nature of quadratic functions and their graphs.

Square Roots

Square roots are fundamental in solving equations like \((3-x)^2 = 25\). When taking square roots, you need to consider both the positive and negative roots, as squaring either will give you the original value.

• Taking the square root of both sides of an equation is a common method to simplify and find possible solutions.
• For \((3-x)^2 = 25\), taking the square root of both sides gives two possible equations: \(3-x = 5\) and \(3-x = -5\).

This is because when you square a number, i.e., multiply it by itself, both the positive and the negative values give the same result.
Thus, considering both potential square roots is crucial for not missing solutions. This principle is vital not only in quadratic equations but also in understanding algebra more broadly.

Parabolas

A parabola is the typical shape of a quadratic equation when graphed. In the case of the equation \((3-x)^2 = 25\), the function \(y = (3-x)^2\) creates a U-shaped curve.

• This particular parabola opens upwards, indicating that as you move away from the vertex along the x-axis, the y-values increase.
• The vertex of our parabola is located at \(x = 3\), where the value of \(y\) is at its minimum (zero in this setup).

Understanding the structure of a parabola—how it opens and where its vertex lies—provides crucial insights into the function’s behavior and helps in identifying solutions graphically.
Moreover, knowing that the equation takes this geometric form helps in understanding why and how quadratic equations can possess two solutions. The symmetric nature of parabolas means intersections with a horizontal line, such as \(y = 25\), occur consistently on both sides of the vertex.