Problem 56 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to find all real solutions. You may need to adjust the window size manually or use the ZOOMFIT feature to get a clear graph. $$\text { Solve } \sqrt{x-1.95}-3.6=-2.5$$

Step-by-Step Solution

Answer

The solution for the equation \( \sqrt{x-1.95}-3.6=-2.5 \) is \( x = 3.16 \)

Step 1: Isolate the square root

The first step is to isolate the square root on one side of the equation. To do this, add 2.5 to both sides of the equation. This results in the following equation: \( \sqrt{x-1.95} = -2.5+3.6=1.1\)

Step 2: Square both sides

Now, square both sides to eliminate the square root. This leads to \( (x-1.95)= 1.1^2 = 1.21 \)

Step 3: Solve for x

Add 1.95 to both sides to solve for x. This results in \( x = 1.21+1.95 = 3.16 \)

Step 4: Check the solution

Now check the solution by substituting x= 3.16 into the original equation. If both sides of the equation are equal, then the solution is correct. If not, adjust the window or use the ZOOMFIT feature on your graphing utility.

Key Concepts

Graphing Utilities

Graphing utilities, such as graphing calculators or software applications, are powerful tools that assist in visualizing mathematical equations and their solutions. In the context of algebraic equations, like the one given in the exercise, graphing utilities allow us to clearly plot complicated functions and discern patterns or points of interest. They are particularly useful for:

• Finding the intersection points which often correspond to the solutions of equations.
• Adjusting the visibility using features like changing window size or ZOOMFIT, enabling an optimized view of the graph.
• Validating the analytic solution derived through algebraic manipulation with a graphical view.

To use a graphing utility effectively, start by entering the function and adjusting settings to ensure that the graph is visible. Analyze the plotted curve to identify where it intersects the x-axis, as these intersections indicate the real solutions.

Square Root Equations

Square root equations involve expressions that contain the square root symbol, often requiring special attention to solve. The primary goal when working with such equations is to isolate the square root expression, providing a clear path to find the solution.

Here's a step-by-step approach:

• Start by isolating the square root expression on one side of the equation. This is essential to simplify and solve the equation effectively. In this exercise, we adjust both sides of the equation to isolate \( \sqrt{x-1.95} \).
• Once isolated, square both sides of the equation to eliminate the square root. This transforms the square root equation into a solvable algebraic equation, which in our example became \( (x-1.95) = 1.21 \).
• Finally, solve for the variable by performing any additional arithmetic operations needed. This involves adding constants or performing operations to isolate \( x \), as demonstrated in the final step of solving \( x = 3.16 \).

These steps must be followed carefully to ensure a successful solution and to prevent errors during the process.

Solution Checking

Solution checking is a crucial step in solving algebraic equations, particularly those involving complex operations like square roots. After calculating a potential solution, it is important to verify its validity by substituting the value back into the original equation.

For our example, the solution \( x = 3.16 \) was verified by plugging it back into the original equation. After substitution:

• If the left-hand side equals the right-hand side of the equation consistently, the solution is confirmed correct.
• If the values don’t match, it indicates a mistake in the calculation or a need to readjust the constraints or settings on the graphing utility (if used).

Solution checking serves as a valuable safeguard in mathematical problem-solving, ensuring reliability and accuracy of the results obtained.