Solve the given equation or inequality graphically.
(a) $x^3-4 x-1=0$
(Enter your answers as a comma-separated list. Round your answers to two decimal places.)
Answer (1)
Analyze the equation x 3 − 4 x − 1 = 0 and consider the function f ( x ) = x 3 − 4 x − 1 .
Graph the function to find where it intersects the x-axis.
Approximate the x-coordinates of the intersection points.
The solutions, rounded to two decimal places, are − 1.86 , − 0.25 , 2.11 .
Explanation
Understanding the Problem We are given the equation x 3 − 4 x − 1 = 0 and asked to solve it graphically. This means we need to find the points where the graph of the function f ( x ) = x 3 − 4 x − 1 intersects the x-axis. These intersection points represent the real roots of the equation.
Graphical Approach To find the roots graphically, we can use a graphing tool or software to plot the function f ( x ) = x 3 − 4 x − 1 . By observing the graph, we can identify the points where the curve crosses the x-axis. These points are the solutions to the equation.
Finding the Roots Using a calculator or software, we find the approximate real roots of the equation x 3 − 4 x − 1 = 0 are approximately -1.86, -0.25, and 2.11. We can verify these roots by plugging them back into the equation to see if they result in a value close to zero.
Final Answer Rounding the roots to two decimal places, we have x ≈ − 1.86 , − 0.25 , 2.11 . These are the x-coordinates where the graph of f ( x ) = x 3 − 4 x − 1 intersects the x-axis.
Examples
Consider designing a bridge where the curve of the supporting arch is modeled by a cubic equation. Finding the roots of this equation helps engineers determine the points where the arch meets the ground. Similarly, in physics, if you're analyzing the trajectory of a projectile and the height is described by a cubic function, finding the roots tells you when the projectile hits the ground. This problem demonstrates how solving cubic equations can be applied in real-world engineering and physics scenarios to find critical intersection points.
