Problem 1 - Beginning and Intermediate Algebra

Question

An equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are real numbers and \(a \neq 0,\) is \(\mathrm{a}(\mathrm{n})\) _________ equation, also called a(n) ________ degree equation. The greatest degree of the variable is _____.

Step-by-Step Solution

Answer

quadratic, second, 2

Step 1: Identify the Form of the Equation

The given equation is of the form: \(ax^2 + bx + c = 0\) where \(a, b,\) and \(c\) are real numbers and \(a eq 0\).

Step 2: Determine the Name of the Equation

An equation of the form \(ax^2 + bx + c = 0\) is known as a quadratic equation.

Step 3: Identify the Degree of the Equation

For the equation \(ax^2 + bx + c = 0\), the highest power of the variable \(x\) is 2. Therefore, it is also called a second degree equation.

Step 4: Determine the Greatest Degree of the Variable

In the quadratic equation \(ax^2 + bx + c = 0\), the greatest degree of the variable \(x\) is 2.

Key Concepts

headline of the respective core concept

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are real numbers and \( a eq 0 \). This type of equation is also referred to as a second degree equation because the highest power of the variable \( x \) is 2. Understanding these equations is essential because they frequently appear in various areas of math and science. Quadratic equations can model a wide range of phenomena, from projectile motion to market economics.

It's crucial to recognize a few key aspects:

• The term \( ax^2 \) is the quadratic term because it includes \( x \) raised to the second power.
• The term \( bx \) is the linear term because it includes \( x \) raised to the first power.
• The term \( c \) is the constant term since it does not involve the variable \( x \) at all.

These terms together define the central characteristics of quadratic equations.

second degree equation

A second degree equation is another name for a quadratic equation. It gets this name because the highest power of the variable is 2. This might sound a bit technical, but breaking it down will make it clearer.

In math, the 'degree' of an equation is the highest power of the variable in it. For example, in \( x^2 + 5x + 6 \), the highest power of \( x \) is 2, so it’s a second degree equation.

Let’s look at the general form of a quadratic equation again: \( ax^2 + bx + c = 0 \). The term \( ax^2 \) is what makes it second degree. Unlike first-degree (linear) equations—like \( 2x + 3 = 0 \)—these equations may have up to two solutions. These solutions can be found using various methods such as factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Understanding the degree helps in identifying the methods to solve it.

headline of the respective core concept

The terms of a polynomial, like those in a quadratic equation, are the different parts that are added together. In the equation \( ax^2 + bx + c = 0 \), we have three terms:

• \( ax^2 \): This is called the quadratic term. It's the term with the highest power of \( x \) which is the second power.
• \( bx \): This is the linear term. It's called 'linear' because the highest power of \( x \) here is one.
• \( c \): This is the constant term. This term doesn’t have \( x \) in it at all; it’s a constant.

Understanding these terms is vital because they tell us about the shape and position of the graph of the equation. For example, changing \( a \) can make the parabola wider or narrower, while changing \( c \) shifts it up or down. Each term plays an essential role in influencing the graph and the solutions of the quadratic equation.