Question

a) Determine the roots of the rational equation \(-\frac{2}{x}+x+1=0\) algebraically. b) Graph the rational function \(y=-\frac{2}{x}+x+1\) and determine the \(x\) -intercepts. c) Explain the connection between the roots of the equation and the \(x\) -intercepts of the graph of the function.

Short answer

The roots, \(x=1\) and \(x=-2\), are also the x-intercepts of the function's graph.

Step-by-step solution

Move all terms to one side

First, rewrite the equation \(-\frac{2}{x}+x+1=0\) with all terms on one side: \(-\frac{2}{x} + x + 1 = 0 \).

Multiply through by the common denominator

Multiply every term by the common denominator, which is \(x\): \(-2 + x^2 + x=0 \).

Rearrange into standard quadratic form

Rewrite the equation in standard quadratic form: \(x^2 + x - 2 = 0 \).

Solve the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) where \(a=1, b=1, c=-2\): \(x = \frac{-1 \pm \sqrt{(1)^2-4(1)(-2)}}{2(1)} = \frac{-1 \pm \sqrt{1+8}}{2} = \frac{-1 \pm 3}{2}\). So, \(x=1\) and \(x=-2\).

Graph the function

Plot the function \(y = -\frac{2}{x} + x + 1\) on a Cartesian plane. Identify the points where the graph touches or crosses the x-axis. These points are the solutions we found in step 4: \(x=1\) and \(x=-2\).

Determine the x-intercepts

From the graph, observe that the x-intercepts are at \(x=1\) and \(x=-2\). These are the points where the function \(y\) equals zero.

Explain the connection

The roots of the equation \(-\frac{2}{x} + x + 1 = 0\) are the same as the x-intercepts of the function \(y=-\frac{2}{x}+x+1\). This is because the roots of the equation represent the values of \(x\) where the function equals zero, which corresponds to the x-intercepts on the graph.

Key concepts

Roots of equations

To solve an equation, you need to find the roots, which are the values of the variable that make the equation true. For the equation \(-\frac{2}{x}+x+1=0\), the roots are the values of \(x\) that satisfy this condition. Think of it as the points where the function touches or crosses the x-axis, making the equation equal to zero. Here, we moved all terms to one side and simplified the equation to \(x^2 + x - 2 = 0\) to solve it more easily.

Quadratic formula

The quadratic formula is a critical tool for solving quadratic equations like \(x^2 + x - 2 = 0\). The formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^2 + bx + c = 0\). Plugging in our values: \(a=1\), \(b=1\), and \(c=-2\), we get \(x = \frac{-1 \pm \sqrt{1+8}}{2} = \frac{-1 \pm 3}{2}\). This gives us two roots, \(x=1\) and \(x=-2\). These roots are essential for graphing and further analysis.

Graphing rational functions

Graphing the function \(y = -\frac{2}{x} + x + 1\) helps visualize where the function intersects the x-axis. These intersection points correspond to the roots of the equation, meaning where \(y = 0\). When graphing, plot several values and connect the points smoothly. Notice how the graph behaves near the x-intercepts \(x=1\) and \(x=-2\). This visual aid is beneficial for understanding the function's behavior and verifying the algebraic solutions.

x-intercepts

The \(x\)-intercepts of a function are the points where the graph crosses the x-axis. For the function \(y = -\frac{2}{x} + x + 1\), the \(x\)-intercepts are at \(x=1\) and \(x=-2\). These intercepts confirm the roots we found using the quadratic formula. The \(x\)-intercepts represent the values of \(x\) that make the output, or \(y\)-value, zero. Check these points by setting \(y = 0\) and solving the resulting equation to find the same roots.

Solving algebraic equations

Solving algebraic equations involves finding the variable values that satisfy the equation. For our example, after rewriting the equation \(-\frac{2}{x} + x + 1 = 0\), simplify it to a quadratic form, \(x^2 + x - 2 = 0\). Use techniques like the quadratic formula to find solutions. Always simplify and verify your solutions by plugging them back into the original equation. Each root reveals important characteristics about the graph of the corresponding function.