Problem 64 - Algebra 1

Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{15-4 x}=2 x$$

Step-by-Step Solution

Answer

The solutions are usually dependent on how accurately you capture the intersection points in your graphing calculator so an exact answer cannot be provided. Analytically validate your solution by substituting your \(x\) value(s) back into the original equation.

Step 1: Graph a function

Enter the function \(y=\sqrt{15-4 x}\) into your graphing calculator. This will give you the graph of the left side of the equation.

Step 2: Graph another function

Next, enter the function \(y=2x\) into your graphing calculator. This will give you the graph of the right side of the equation.

Step 3: Find the intersection

Now, find the x-values at which these two functions intersect. This will be your solutions graphically.

Step 4: Check the solution algebraically

To confirm the solution, replace \(x\) in the equation \(\sqrt{15-4x}=2x\) with the values found in Step 3. If the left and right sides of the equation are equal, then the solutions are correct.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize mathematical equations. It is essential in graphing functions and finding their points of intersection, especially when a problem involves complex operations like square roots or radicals.

For the equation \(\sqrt{15-4x}=2x\), the first step is to input the functions separately into the graphing calculator:

• Enter \(y = \sqrt{15-4x}\) - This represents one side of the radical equation.
• Enter \(y = 2x\) - This represents the other side of the equation.

The calculator will display the graphs of both functions on the same set of axes, making it easier to locate where they intersect. Graphing calculators allow you to zoom in or out and pan around graphs to better see the curves and their intersections.

With these functionalities, graphing calculators prove to be invaluable for students tackling equations that are otherwise challenging to solve manually.

Intersection of Functions

The intersection of functions refers to the point or set of points where the graphs of two functions meet. In terms of solving equations, these intersections represent the solution(s) to the equation. Understanding this visually is crucial because it shows where the values of each function are equal.

In the equation \(\sqrt{15-4x} = 2x\), after graphing the functions \(y = \sqrt{15-4x}\) and \(y = 2x\), you can see where they cross.

• The x-coordinate(s) of the intersection point(s) are solutions to the equation.
• This is because the point of intersection signifies that both sides of the original equation give the same output.

Finding intersections graphically provides not only a solution but also an intuitive understanding of how the functions behave relative to each other. It's a powerful visual aid that makes understanding roots of equations much more tangible.

Algebraic Verification

Once you've determined the solution graphically by finding where the two functions intersect, it's important to verify the solution algebraically. This ensures that the values obtained are correct and meet the mathematical criteria.

For the equation \(\sqrt{15-4x} = 2x\), substitute the x-value of the intersection point back into the original equation. If you've correctly identified the intersection point:

• The left side, \(\sqrt{15-4x}\), yields a number.
• The right side, \(2x\), should yield the same number.

If both sides of the equation are equal upon substitution, then the solution is verified. Algebraic verification is crucial as it serves as a double-check, ensuring no graphical or computational errors led to a misinterpretation.