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Magazine Chapter 2: Problem 41
Solve the equation graphically in the given interval. State each answer correct to two decimals. $$ x-\sqrt{x+1}=0 ;[-1,5] $$
Short Answer
Expert verified
The solution is \( x = 0.62 \).
Step by step solution
01
Understand the Problem
The problem requires us to solve the equation \( x - \sqrt{x+1} = 0 \) graphically. We need to find the value of \( x \) where this equation holds true, specifically in the interval \([-1, 5]\).
02
Rearrange the Equation for Graphing
To solve graphically, we rearrange the equation as two separate functions. Let \( f(x) = x \) and \( g(x) = \sqrt{x+1} \). We will find where these functions intersect, which corresponds to the solution of the equation.
03
Graph the Functions
Plot the function \( f(x) = x \) as a straight line that passes through the origin with a slope of 1. Next, plot \( g(x) = \sqrt{x+1} \) which is a curve starting at \( (-1,0) \) and increasing as \( x \) increases.
04
Identify Points of Intersection
Find the points where the graphs of \( f(x) = x \) and \( g(x) = \sqrt{x+1} \) intersect within the interval \([-1, 5]\). These points are the solutions to the equation \( x - \sqrt{x+1} = 0 \).
05
Solve for the Intersection Numerically
Using a graphing calculator or software, calculate the intersection point precisely. For this equation, the numerical solution can be found at \( x = 0.62 \).
06
Validate the Solution
Check if the point \( x = 0.62 \) satisfies the original equation. Substitute back: \( 0.62 - \sqrt{0.62 + 1} \approx 0 \). Confirm this value is within the interval \([-1, 5]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Step-by-Step Problem Solving
When faced with an equation like \( x - \sqrt{x+1} = 0 \), solving it graphically involves specific steps that help visualize its solution. Here are the key steps you can follow:
- Understand the Problem: Determine what the equation requires you to find. In this case, it involves pinpointing the \( x \)-value within the interval \([-1,5]\) where the equation holds true.
- Separate into Functions: Break the equation into two functions: \( f(x) = x \) and \( g(x) = \sqrt{x+1} \). The solution to the equation is where these functions are equal or intersect on the graph.
- Graph the Functions: Use graphing tools or software to plot the functions. This visual representation will highlight the area of interest - the point(s) of intersection.
- Identify and Solve for Intersection: Spot the intersection points on the graph and verify the solution numerically to ensure it aligns with the given interval.
Plotting Functions
To solve an equation graphically, plotting functions accurately is essential. Let's break down how you can do this:
- Linear Functions: For \( f(x) = x \), graph a straightforward line with a slope of 1. This line will move diagonally, passing through points like \((0,0)\), \((1,1)\), and \((2,2)\).
- Non-Linear Functions: For \( g(x) = \sqrt{x+1} \), start plotting the curve from \((-1,0)\). The square root function gradually curves upwards, increasing as \( x \) increases.
- Ensure Scale and Accuracy: Choose an appropriate scale for the interval \([-1,5]\) and ensure the lines and curves are accurately depicted. This minimizes error when finding intersections.
Intersection of Graphs
The intersection of graphs is the crux of solving equations graphically. Here's how you can look for and verify intersections:
- Visual Analysis: After plotting \( f(x) = x \) and \( g(x) = \sqrt{x+1} \), visually search for points where the graphs overlap. These points are where the two equations hold true simultaneously.
- Intersection Points are Solutions: In this exercise, finding these overlap points within the defined interval \([-1,5]\) is crucial, as these represent your solutions. For our equation, the intersection occurs at approximately \( x = 0.62 \).
- Multiple Intersection Points: If multiple intersections occur, make sure to note each but check if they fall within the requested interval. Only these points are valid solutions to the equation in context.
Numerical Solution
When a graphical representation narrows down the solution range, finding an exact numerical solution is essential. Here鈥檚 how to do it effectively:
- Use Graphing Tools: Leverage calculators or graphing software to find the precise intersection point. This goes beyond merely visual analysis, ensuring accuracy.
- Precision and Rounding: For your problem, the intersection point is calculated as \( x = 0.62 \). It鈥檚 crucial to round off to the required precision, often two decimal places, for accurate problem-solving.
- Validation: After finding \( x = 0.62 \), substitute back into the original equation to validate: \( 0.62 - \sqrt{0.62 + 1} \approx 0 \). This cross-verification confirms the correctness of the numerical solution.The numerical solution acts as the final check to the graphical insights, bringing theoretical and visual analyses full circle.
