Problem 56 - Algebra 1

Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{x+4}=3$$

Step-by-Step Solution

Answer

From the graphical solution on the calculator, the answer to the equation is \(x = 5\). This can be validated by substitifying back into the equation: \( \sqrt{5 + 4} = 3\), which simplifies to \(3 = 3\).

Step 1: Solve Graphically

To solve graphically using a graphing calculator, plot two functions: \(y = \sqrt{x + 4}\) and \(y = 3\). Where the two graphs intersect on the x-axis are the solutions.

Step 2: Discover Intersection Point

Find the point of intersection of the two curves. This can usually be done by using the 'intersect' function in the graphing calculator. The x-coordinate of that intersection point is the solution to the equation.

Step 3: Verify Algebraically

To check the solution algebraically, substitute the x-value found in step 2 back into the original equation. Both sides of the equation should balance, proving the validity of the solution.

Key Concepts

Graphing Calculator

The graphing calculator is an indispensable tool for visualizing math problems, especially when it comes solving radical equations like \( \sqrt{x+4}=3 \). These calculators allow students to plot equations and find points of intersection quickly. To utilize a graphing calculator effectively:

• Enter the radical equation by breaking it down into two parts that can be graphed: \(y = \sqrt{x + 4}\) and \(y = 3\).
• Use the plot or graph function to visualize these equations on the same coordinate plane.
• The 'intersect' function can be employed to pinpoint where the graphs cross paths, which corresponds to the solution of the equation.

This visual approach complements algebraic methods and can provide immediate, intuitive insight into the behavior of the functions involved.

Radical Equation

A radical equation is one that includes a variable within a radical, typically a square root. In the equation \( \sqrt{x+4}=3 \), the radical \( \sqrt{} \) signifies the principal square root. Solving these equations often requires isolating the radical on one side of the equation and then squaring both sides to eliminate it.

• Ensure that all possible solutions are checked, as squaring both sides of an equation can introduce extraneous solutions.
• When solutions are found, they must be substituted back into the original equation for verification.

This helps in ensuring that the solutions found are valid within the context of the equation.

Graphical Solution

Graphical solutions are a way to solve equations by interpreting the points of intersection on a graph. When we plot two functions from a radical equation, their points of intersection represent the solution to that equation. For instance:

• By graphing \(y = \sqrt{x + 4}\) alongside \(y = 3\), we are setting up a scenario whereby the 'x' values at which these two graphs meet, represent the solution(s) to the equation \( \sqrt{x+4}=3 \).
• This method not only offers a visual representation of the solution but also makes comprehending the range and domain of the functions easier.

A graphical solution can be remarkably intuitive as it illustrates how the functions behave in relation to each other over a range of values.

Algebraic Verification

Algebraic verification is the process of confirming that the solutions obtained through graphical or other methods actually satisfy the original equation. It acts as a check against potential errors made while solving the equation visually or computationally. Here's how to perform an algebraic verification:

• Start with the potential solution from the graph, for instance, the 'x' value derived from the intersection of \(y = \sqrt{x + 4}\) and \(y = 3\).
• Substitute this 'x' value back into the original radical equation to see if the right side equals the left side.
• Ensure that both sides of the equation are indeed equal, confirming the solution is correct.

This step is necessary to rule out the possibility of extraneous solutions, which can arise due to the nature of radical equations and the methods employed to solve them.