Problem 55 - Brief Applied Calculus

Question

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (Round answers to two decimal places.) $$ 2 x^{2}+40=18 x $$

Step-by-Step Solution

Answer

The solutions to the equation are approximately x = 4.44 and x = 4.56.

Step 1 - Set the Equation

First, rearrange the given equation into a standard form. The equation given is \(2x^2 + 40 = 18x\). Move all terms to one side to set the equation to \(0\): \[ 2x^2 - 18x + 40 = 0 \]

Step 2 - Graph the Equation

Now, use a graphing calculator to graph the equation \(y = 2x^2 - 18x + 40\). Set the window to \([-10, 10]\) for both x and y-axes, which provides a view of the potential x-intercepts.

Step 3 - Identify the Zeros

With the graph displayed, use the ZERO or TRACE function of the calculator to find where the graph crosses the x-axis. These points are the solutions to the equation \(2x^2 - 18x + 40 = 0\).

Step 4 - Refine the Solution

Utilize the ZOOM IN feature if necessary to get a more precise view of where the graph intersects the x-axis. Check these x-values, rounding them to two decimal places.

Step 5 - Record the Solutions

After rounding the x-intercepts found by the calculator, record these values as the solutions to the original equation. You should find two solutions.

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly useful tool for solving quadratic equations, such as the one given in our problem. Unlike a basic calculator that only handles simple arithmetic, a graphing calculator can plot graphs of equations, giving you a visual representation of the solutions. In the case of quadratic equations, this means you can see the characteristic U-shaped curve, known as a parabola.

To solve the given equation, the first step is to rearrange it into a standard form and then input it into the calculator. The equation becomes y = 2x² - 18x + 40. By graphing this function and adjusting the window to egin{align*} [-10, 10] ext{ for both the } ext{x and y-axes,} ext{ you can } ext{adequately sight where } ext{the parabola intersects the x-axis.} ext{ These intersection points, known as } ext{x-intercepts, are the solutions to the quadratic equation.} Utilize the ZERO, TRACE, or ZOOM IN features to pinpoint these solutions accurately.

• ZERO Function: Find where the graph crosses the x-axis.
• TRACE Function: Move along the graph to find approximate intersection points.
• ZOOM IN: Get a closer look at the specific point where the graph meets the x-axis.

This process provides a calculative and visual method to find solutions, offering a more comprehensive understanding.

Quadratic Formula

The quadratic formula is another powerful tool for solving quadratic equations. Unlike a graphing calculator, which provides a visual solution, the quadratic formula gives a precise numerical result through algebraic manipulation. The general form of a quadratic equation is \(a x^{2} + b x + c = 0\), and the solutions are given by the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the coefficients from your rearranged equation (a = 2, b = -18, and c = 40) into the formula to find the exact solutions for \(x\). The calculations will include finding the discriminant \(b^2 - 4ac\), which indicates whether the solutions are real and distinct, real and repeated, or complex.

• The Discriminant: A positive discriminant implies two distinct real solutions. A zero discriminant means one real, repeated solution. A negative discriminant indicates complex solutions.

Using the quadratic formula provides an exact solution to any quadratic equation, regardless of the position of any graph, and confirms the results you might visually gauge from a graphing calculator.

Parabola

A parabola is a symmetrical plane curve formed by graphing a quadratic equation. Its distinctive "U-shape" is determined by the equation \(y = ax^2 + bx + c\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

The central point of a parabola is its vertex, which is its highest or lowest point depending on the orientation. Along the horizontal line known as the x-axis, the parabola's mirror-like symmetry can be observed.

• Vertex: The vertex can be found using \(x = \frac{-b}{2a}\), where \(b\) and \(a\) are from the quadratic equation's coefficients.
• Axis of Symmetry: A vertical line passing through the vertex, defining the parabola's symmetry.
• Direction: Determined by the sign of \(a\) in the equation.

Understanding the basic structure of a parabola helps in using tools like a graphing calculator, as it underscores where critical points such as x-intercepts might be found. This insight is crucial for solving quadratic equations graphically.

X-Intercepts

In a quadratic equation graph, the x-intercepts are the points where the parabola crosses the x-axis. These represent the solutions to the quadratic equation. For the equation $y = 2x^2 - 18x + 40$, x-intercepts correspond to the values of $x$ that satisfy $y=0$.

• Finding X-Intercepts: Usually achieved through setting $y = 0$ in the quadratic equation and solving for $x$ either through factored form, using the quadratic formula, or observing on a graph.
• Visualizing: On a graphing calculator, these are visible as the points where the parabola touches or intersects the x-axis.
• Number of Intercepts: A quadratic equation can have two, one, or no x-intercepts depending on the value of the discriminant.

Identifying the x-intercepts is vital since they give the exact solutions of the quadratic equation reflecting where the function depicted by the graph equals zero. Using either a graphing calculator or algebraic methods helps to pinpoint these critical solutions effectively.