Problem 124 - Algebra and Trigonometry

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

Solving the equation involving a square root function graphically consists of graphing the function within the given viewing rectangle, determining the x-intercepts of the curve, and confirming these solutions through substitution into the original equation. The exact solutions depend on the graph and can be precisely realized through a graphing utility.

Step 1: Plot the Function in the Viewing Rectangle

Plot the square root function \(y = \sqrt{2x + 13} - x - 5\) in the given viewing rectangle [-5, 5, 1] by [-5, 5, 1], which means \(x\) values are in the range of -5 to 5 with increment of 1, and the same for \(y\) values. You could use any graphing tool to generate this plot, where \(y = \sqrt{2x + 13} - x - 5\) will be depicted as a curve on the coordinate plane.

Step 2: Find the X-Intercepts

The x-intercepts are the points on the graph where the curve crosses the x-axis. In other words, they're the \(x\) values when \(y = 0\). Locate these points and record their \(x\) values. They should be the solutions to the equation.

Step 3: Check by Direct Substitution

Take the x-values found from the previous step, and substitute them back into the original equation \(y = \sqrt{2x + 13} - x - 5\), each in place of \(x\) respectively. If the equation holds true and \(y = 0\), then the values are verified as correct. If not, review the previous steps to find and correct any mistakes.

Key Concepts

Understanding X-Intercepts

When graphing equations, one crucial concept is the x-intercept. The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the y-value is zero. In simple terms, we are looking for the x-values that make the equation equal to zero.
For example, if we have an equation like \( y = \sqrt{2x + 13} - x - 5 \), we're interested in finding those points where \( y = 0 \).

• To find the x-intercepts, plot the function on a coordinate plane.
• Look at the points where the curve crosses the x-axis.
• Record the x-values of those points; these are your solutions.

Understanding x-intercepts is helpful because they are the solutions to the equation when it's set equal to zero. It's often easier to find these points graphically, especially when equations are complex and involve radicals or other intricate expressions.

Direct Substitution Method

The direct substitution method is a straightforward way to verify if your solution is correct. This involves taking the x-values you obtained (your potential solutions) and substituting them back into the original equation. In our example, if you found an x-intercept at some point, say \( x = a \), substitute \( a \) back into the equation:
\[ y = \sqrt{2a + 13} - a - 5 \]

• Calculate the value on the right-hand side.
• If the result is equal to zero, your solution is correct.
• If not, recheck your graph or calculations for errors.

This method checks the accuracy of your solutions, providing certainty that the points found on the graph correspond to actual solutions of the equation.

The Square Root Function

The square root function is a type of radical function and is generally written as \( y = \sqrt{x} \). Understanding this function is important as it frequently appears in various mathematical contexts.
In the equation \( y = \sqrt{2x + 13} - x - 5 \), the square root function portion \( \sqrt{2x + 13} \) determines the shape of the curve.

• The domain must ensure that whatever's inside the square root isn’t negative. For \( \sqrt{2x + 13} \), ensure \( 2x + 13 \geq 0 \).
• This means \( x \geq -6.5 \), giving us a valid region to plot the graph.
• The graph will usually start at some point on the x-axis and rise or fall depending on the equation's form.

Understanding these characteristics helps you plot and interpret graphs involving square roots, ensuring you recognize potential restrictions, and how they affect the graph.