Problem 103 - College Algebra

Question

use a graphing utility and the graph's \(x\)-intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$ \begin{aligned} &\sqrt{2 x+13}-x-5=0\\\ &[-5,5,1] \text { by }[-5,5,1] \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the equation involves graphing the function, identifying the x-intercepts and check them by substituting these values back into the original equation.

Step 1: Graph the function

First plot the function \( \sqrt{2 x+13}-x-5 =0 \) on a graph using the given viewing window, [-5,5,1] by [-5,5,1]. The x-axis values range from -5 to 5 and y-axis values also range from -5 to 5.

Step 2: Identify the x-intercepts

The x-intercepts of the graph are the points where the graph crosses or touches the x-axis. Observe where the plot of the function intersects with the x-axis. These points are the solutions of the equation.

Step 3: Direct Substitution

Take the x-coordinates of these intersection points and substitute them back into the original equation \( \sqrt{2 x+13}-x-5 =0 \) . If the left-hand side of the equation equals the right-hand side after the substitution, then the solution is correct.

Key Concepts

x-intercepts

The x-intercepts are a crucial concept when solving equations using graphs. They represent the points where the graph of the function crosses or touches the x-axis. When dealing with the equation \(\sqrt{2x+13} - x - 5 = 0\), the x-intercepts are the values of \(x\) for which the equation equals zero.
To find them, plot the function on a graph and observe where it intersects the x-axis. These intersection points are essentially the solutions to the equation.
Understanding x-intercepts helps us solve equations graphically and offers a visual proof of the solution.

viewing window

A viewing window is essential for graphing functions accurately as it specifies the range of x and y values displayed on the graph. In the exercise, the given viewing window is \([-5, 5, 1]\) by \([-5, 5, 1]\), which means:

• The x-axis ranges from -5 to 5 with a scale or increment of 1.
• The y-axis ranges from -5 to 5 with a scale or increment of 1 as well.

This setting helps us to see the important features of the graph, such as the x-intercepts, clearly and ensures that the points of intersection with the axes are visible. Choosing an appropriate viewing window is critical so you don't miss these features.

direct substitution

Direct substitution is a method used to verify solutions obtained from a graph. Once you have identified potential solutions by finding the x-intercepts, you need to check that these values satisfy the original equation.
To perform direct substitution, take the x-values from the x-intercepts and plug them back into the equation \(\sqrt{2x + 13} - x - 5 = 0\). For instance, if an x-intercept is \(x = a\), substitute \(a\) into the equation and check if both sides equal zero.
If they do, the solution is confirmed as correct. This step ensures accuracy and correctness in the solution process.

square root function

The square root function is an important type of function that has unique characteristics. In the equation \(\sqrt{2x+13} - x - 5 = 0\), \(\sqrt{2x + 13}\) represents the square root function part.
Square root functions typically start at a point where their argument is zero, creating a curved branch that extends to the right for positive x values.

• To understand its behavior, consider that the square root function is only defined when the expression inside the square root is non-negative.
• Hence, for \(\sqrt{2x + 13}\), ensure \(2x + 13 \geq 0\), meaning \(x \geq -6.5\).

This understanding helps when graphing the function and analyzing its effects on the equation's solution.