Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{x-5.6}=2.5$$

Short answer

The solution to the equation \( \sqrt{x-5.6}=2.5 \) can be determined graphically using a calculator and then confirmed algebraically by substituting the solution back into the original equation.

Step-by-step solution

Graphing the equation

Enter the equation \( \sqrt{x-5.6}=2.5 \) into the graphing calculator and plot the graph.

Identify the point of intersection

Find the point where the curve of the function crosses the y-value of 2.5. The x-coordinate of this point is the solution to the equation.

Check the solution algebraically

Take the identified solution and substitute it back into the original equation. If both sides of the equation are equal, then the identified solution is correct.

Key concepts

Graphing Calculator

Graphing calculators are powerful tools that significantly aid in visualizing mathematical problems, especially when working with equations involving radicals. When tasked with graphing a radical equation like \( \sqrt{x-5.6}=2.5 \), a graphing calculator helps to provide a visual representation of the equation's curve. By plotting this on the graphing calculator, students can immediately see the shape of the radical function and understand how it behaves.

To graph the equation, you'll need to input the function into the calculator. This usually involves navigating to the graphing feature, entering the equation exactly as it appears, and then observing the plotted graph on the screen. The curve that represents the equation will help you locate the solution as the point where it intersects the horizontal line where the y-value is 2.5.

A useful tip is to ensure that your graphing window is properly set to include the relevant values. For the equation \( \sqrt{x-5.6} = 2.5 \), setting the window to show x-values starting slightly before 5.6 and above would ensure the relevant portion of the curve is visible. It's this intersection between visualization and calculation that makes using a graphing calculator so beneficial.

Solve Radical Equations

Solving radical equations involves finding the value(s) of the variable that make the equation true. In the case of our example, \( \sqrt{x-5.6} = 2.5 \), we aim to find the 'x' value that, when substituted into the equation, equates the radical expression to 2.5. To solve such an equation algebraically, you would typically isolate the radical on one side and then square both sides to eliminate the radical. This transformed equation can be solved like any other algebraic equation.

In this process, it's crucial to consider the domain of the radical function—since square roots are not defined for negative numbers, any values of 'x' that result in a negative number under the radical sign cannot be considered solutions. Additionally, when you square both sides of an equation, you may introduce extraneous solutions that do not actually satisfy the original equation. Hence, it is important to verify any potential solutions.

Understanding the characteristics of radical functions—their domain restrictions, and how they graphically represent on a calculator—enhances your ability to solve these equations. Practice and familiarity with different types of radical equations can also build confidence in your solving skills.

Algebraic Verification

Algebraic verification is an essential step to confirm that the solutions found by a graphing calculator, or obtained through algebraic manipulation, are indeed accurate. To verify solutions algebraically, you substitute the solution back into the original equation. For the given equation \( \sqrt{x-5.6} = 2.5 \), once you've determined a potential 'x' value through graphing or calculation, you would replace 'x' in the equation with this value.

If both sides of the equation are equal after substitution, then the solution is correct. For instance, if we find that 'x' equals a certain number by using our graphing calculator, plugging this number back in place of 'x' should yield an equality that holds true.

Why is verification so important? It helps eliminate any extraneous solutions that might have been introduced during the process, especially when squaring both sides of an equation. Remember, due to the nature of radicals and the operations performed on them, not all algebraically determined solutions may be valid. Therefore, always perform this crucial check to ensure accuracy, which can save you from potential errors and solidify your understanding of the underlying concepts.