Problem 43 - College Algebra with Modeling and Visualization

Question

Solve the quadratic equation graphically. $$ 20 x^{2}+11 x=3 $$

Step-by-Step Solution

Answer

The solutions are the x-values where the parabola intersects the x-axis.

Step 1: Write the Equation in Standard Form

First, rearrange the given equation \(20x^{2} + 11x = 3\) to standard quadratic form \(ax^2 + bx + c = 0\). To do this, subtract 3 from both sides to get: \[20x^{2} + 11x - 3 = 0\].

Step 2: Identify Coefficients

Identify the coefficients from the equation \(20x^2 + 11x - 3 = 0\). Here, \(a = 20\), \(b = 11\), and \(c = -3\).

Step 3: Sketch the Parabola

Plot the quadratic function \(y = 20x^2 + 11x - 3\) on a graph. This parabola opens upwards since the coefficient of \(x^2\) is positive. The vertex, y-intercept, and symmetry will help define the graph.

Step 4: Determine the Roots Graphically

Find the points where the parabola crosses the x-axis. These x-values are the roots or solutions to the equation \(20x^2 + 11x - 3 = 0\). Use a suitable graphing tool or carefully plotted graph to find where the parabola intersects the x-axis.

Key Concepts

Graphing Parabolas

One of the key steps in solving a quadratic equation graphically is to understand how to graph a parabola. A parabola is a symmetrical, U-shaped curve that can open upwards or downwards, depending on the coefficient of the squared term. If the coefficient (known as 'a') is positive, then the parabola opens upwards; if it is negative, the parabola opens downwards.
To graph the quadratic equation, you need to plot significant points:

• **Vertex**: This is the turning point of the parabola. It can be found using the formula \(x = -\frac{b}{2a}\), and then substituting that back into the equation to find the y-value.
• **Axis of Symmetry**: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• **Intercepts**: The y-intercept is where the parabola crosses the y-axis, found by setting \(x = 0\). The x-intercepts (roots) are where the parabola crosses the x-axis, which are the solutions to the equation.

When graphing, make sure to plot a few additional points on both sides of the vertex for an accurate depiction of the parabolic shape.

Solving Quadratic Equations

Quadratic equations can be solved using several methods, such as factoring, completing the square, using the quadratic formula, or graphically. Solving graphically, as in this example, involves finding where a parabola meets the x-axis.
Graphically, the solutions to a quadratic equation are the x-coordinates of the points where the parabola intersects the x-axis:

• **Zero points**: These are also known as the roots or solutions of the equation. They can be real or complex numbers, although graphically, we mainly identify real solutions.
• **Nature of roots**: If the parabola crosses the x-axis twice, there are two distinct real solutions. If it touches once, there is one real solution (a repeated root). If it does not touch at all, there are no real solutions were graphically visible.
• **Tools**: Using a graphing calculator or software can greatly help in precisely identifying these points of intersection.

Understanding how the graph represents the equation’s solutions is an essential concept in the study of quadratic equations.

Standard Form of Quadratic Equations

The standard form of a quadratic equation is given by \(ax^2 + bx + c = 0\). It is the most common and useful form for solving, analyzing, and graphing quadratic equations. In this form:

• **"a" coefficient**: Determines the direction and width of the parabola. Larger values create a narrower parabola.
• **"b" coefficient**: Influences the position of the vertex and the axis of symmetry.
• **"c" coefficient**: The y-intercept, which is the point where the parabola crosses the y-axis.

Rewriting equations in standard form ensures they are in a recognizable, organized structure. This makes it easier to apply quadratic solutions techniques like factoring or using the quadratic formula. Standard form also allows for quick identification of key graphing points, making it a critical step in analyzing quadratic functions.