Problem 58 - Algebra 1

Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{9.2-x}=1.8$$

Step-by-Step Solution

Answer

The solution is the \(x\) value at the intersection point of the functions. After checking algebraically by squaring both sides of the equation and solving for \(x\), the answer obtained should match the \(x\) value at which the functions intersect graphically. The graphing calculator makes the process of finding the intersection point much easier.

Step 1: Graphing the function

Enter \(y=\sqrt{9.2-x}\) and \(y=1.8\) into the graphing calculator to graph the functions. Two lines will be generated, one representing the square root function and the other representing the constant function \(y=1.8\). The solution to the problem is the \(x\) value at which these two lines intersect.

Step 2: Finding the intersection point

Using the intersection function on the graphing calculator, determine the \(x\) value at the intersection point to find the solution. This should be the point where the two functions cross each other.

Step 3: Checking the solution algebraically

Square both sides of the equation \(y=\sqrt{9.2-x}=1.8\) to get \(9.2-x = 1.8^2\). Solve this equation to get the value of \(x\), which should match the intersection point found in step 2. Squaring both sides removes the square root, making it easier to solve the equation algebraically.

Key Concepts

Radical Equations

Radical equations are equations that involve a variable within a radical, typically a square root. These equations often require careful manipulation to solve because of the presence of the radical sign. To work through a radical equation, consider these steps:

• Isolate the radical on one side of the equation, if possible.
• Square both sides of the equation to eliminate the square root. This step is crucial as it transforms the equation into a simpler form without radicals.

By eliminating the radical, the equation becomes linear or sometimes quadratic, which are generally easier to handle. However, it's important to remember that squaring both sides may introduce extraneous solutions. Hence, verifying the solution within the context of the original equation is a good practice to ensure accuracy.

Intersection Points

In mathematics, the intersection points between two functions graphically represent the solutions to an equation where the functions are set equal to each other. For example, based on our original exercise, the functions

• \( y = \sqrt{9.2-x} \)
• \( y = 1.8 \)

need to be graphed to find where they intersect. The intersection point occurs where both functions have the same \( y \)-value for a common \( x \)-value. This intersection represents the solution to the equation. By utilizing a graphing calculator, you can visually identify the intersection point which simplifies finding the solution. Graphing provides a clear, visual method to verify that the algebraic form of the equation and the graphical representation converge to the same result.

Algebraic Solutions

After graphically identifying the intersection point, it's crucial to verify the solution algebraically to ensure its legitimacy. Let's walk through verifying a solution algebraically:

• Start with the original equation: \( \sqrt{9.2-x} = 1.8 \).
• Square both sides: This gives \( 9.2-x = 1.8^2 \), resulting in \( 9.2-x = 3.24 \).
• Solve for \( x \): Rearranging gives \( x = 9.2 - 3.24 \).
• Calculate \( x \): \( x = 5.96 \).

This test confirms that the solution is valid because \( x = 5.96 \) matches the value obtained from the graphical method. Checking the solution helps to ensure that any potential errors introduced by squaring are corrected and that the solution satisfies the original radical equation. This two-step confirmation is key to mastering radical equations.