Problem 14 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=144$$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = -12\).

Step 1: Identify the Equation Type

The given equation is \(x^2 = 144\). This is a quadratic equation because it involves a variable, \(x\), raised to the power of 2.

Step 2: Apply Square Root Method

To solve \(x^2 = 144\), take the square root of both sides of the equation. This gives us \(x = \pm\sqrt{144}\).

Step 3: Calculate the Square Root

Calculate \(\sqrt{144}\), which is 12. Thus, \(x = \pm 12\). This means \(x = 12\) or \(x = -12\).

Step 4: Graphical Representation

Graph the function \(y = x^2\) and \(y = 144\). Notice that the graphs intersect at two points: \((12, 144)\) and \((-12, 144)\). These intersections confirm the solutions \(x = 12\) and \(x = -12\).

Key Concepts

Square Root Method

When solving the quadratic equation \(x^2 = 144\), one effective method is the Square Root Method. This technique is often used when the equation is already in the form \(x^2 = k\), where \(k\) is a constant number. Here's how it works:

• Step 1: Isolate the quadratic term. Ensure that the equation is in the form \(x^2 = k\). In our case, the equation is already \(x^2 = 144\).
• Step 2: Apply the square root. To eliminate the square, take the square root of both sides. But remember, the square root method gives two possible solutions since both positive and negative numbers have the same square. It leads to: \[x = \pm\sqrt{144}\]
• Step 3: Simplify the square root. Calculate \(\sqrt{144}\), which is 12. Therefore, \(x = \pm 12\) or \(x = 12\) and \(x = -12\).

By using the square root method, we efficiently find the solutions for quadratic equations where the right-hand side is a perfect square.

Graphical Representation

Graphical Representation is a powerful tool to visually confirm solutions of equations. For the quadratic equation \(x^2 = 144\), we can represent it on a graph to better understand its solutions. Let's explain how this works:

• Step 1: Graph the first function. Plot the function \(y = x^2\) on a coordinate plane. It appears as a parabola opening upwards, with its vertex at the origin (0,0).
• Step 2: Graph the second function. Plot the line \(y = 144\), which is a horizontal straight line passing through the y-axis at 144.
• Step 3: Identify Intersection Points. The points where the parabola and the line intersect represent the solutions to the equation. These intersection points occur at \((12, 144)\) and \((-12, 144)\).

This graphical method allows you to see that the equation has two solutions: \(x = 12\) and \(x = -12\), corresponding to the x-values where the curves intersect. It's a visual check that complements algebraic methods.

Intersection Points

In solving equations graphically, identifying Intersection Points plays a crucial role. These points show where the graphs of two functions meet and highlight the solutions to the equation. For our equation \(x^2 = 144\), let's explore this further:

• Understanding Intersection Points. The intersection points occur when the value of the parabola \(y = x^2\) coincides with the value of the line \(y = 144\). This happens at the x-values that satisfy the equation \(x^2 = 144\).
• Solutions Confirmed. The intersection points found at \((12, 144)\) and \((-12, 144)\) confirm the solutions \(x = 12\) and \(x = -12\) derived from the algebraic method.
• Visual Validation. These points serve as a visual validation that the algebraic methods were correctly applied, ensuring the robustness of the solution process.

Understanding and identifying intersection points not only solidifies your grasp of the equation's solutions but also demonstrates the harmony between algebraic and graphical methods.