Problem 96 - College Algebra

Question

Explain how to recognize an equation that is quadratic in form. Provide two original examples with your explanation.

Step-by-Step Solution

Answer

A quadratic equation can be recognized by the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are numbers and \(a\) is not zero. Two examples of this are: \(2x^2 + 3x - 4 = 0\) and \(5x^2 - 7x + 8 = 0\).

Step 1 - Identify Quadratic Equations

One can recognize quadratic equations by their signature form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are numbers, and \(a\) is not zero. The presence of a squared term and no higher powers identify this equation type.

Step 2 - Outline Notable Features

What characterizes quadratic equations is the highest power, which is two. They contain three terms - the highest power term \(ax^2\), the linear term \(bx\), and the constant term \(c\). Understanding these features will help in recognizing these equations.

Step 3 - Creation of Example 1

Let's generate the first example: \(2x^2 + 3x - 4 = 0\). It's essential to note that it follows the quadratic form \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = 3\), and \(c = -4\).

Step 4 - Creation of Example 2

Here's a second example: \(5x^2 - 7x + 8 = 0\). Again, it adheres strictly to the form of the quadratic equation. In this case, \(a = 5\), \(b = -7\), and \(c = 8\).