Problem 18 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 8 a^{2}-2 a=0 $$

Step-by-Step Solution

Answer

Question: Given the quadratic equation \(8a^2 - 2a = 0\), determine the values of \(a, b,\) and \(c\) for the general form of a quadratic equation, which is \(ax^2 + bx + c = 0\). Answer: For the given quadratic equation \(8a^2 - 2a = 0\), the values of \(a, b,\) and \(c\) are \(a = 8\), \(b = -2\), and \(c = 0\).

Step 1: Identify the terms in the given equation

Let's look at the equation given: \(8a^2 - 2a = 0\) Here, we can see that the terms are \(8a^2\) and \(-2a\).

Step 2: Match the terms with the general form

The general form of a quadratic equation is \(ax^2 + bx + c = 0\). In this case, the variable is \(a\). So, we rewrite the general form as \(ax^2 + bx + c = 0\). Now let's match each term: 1. \(8a^2\) matches the \(ax^2\) term: \(a = 8\). 2. \(-2a\) matches the \(bx\) term: \(b = -2\). 3. There is no constant term in the equation, so: \(c = 0\). So, we have determined the values of \(a\), \(b\), and \(c\): \(a = 8\), \(b = -2\), and \(c = 0\).

Key Concepts

Identifying Coefficients in Quadratic Equations

In any quadratic equation, coefficients are the numbers in front of the variables. They play a crucial role in describing the equation's features. Let's discuss how to identify them using a problem as an example.

Given a quadratic equation like \(8a^2 - 2a = 0\), our task is to determine the coefficients \(a\), \(b\), and \(c\). These are parameters from the standard form of a quadratic equation. In this problem, you'll find coefficients:

• The term \(8a^2\) indicates a coefficient of 8 in front of \(a^2\). Thus, \(a = 8\).
• The term \(-2a\) shows a coefficient of -2 in front of \(a\). Thus, \(b = -2\).
• There is no constant term visible, meaning \(c = 0\).

Once these are identified, they define the equation and allow you to solve it algebraically or analyze its properties.

Understanding Polynomial Terms

In any polynomial equation, such as a quadratic one, understanding its terms is essential. Polynomial terms are individual elements in an equation consisting of a variable raised to a power and a coefficient attached to it. In the quadratic equation \(8a^2 - 2a = 0\), we identify:

• \(8a^2\) - This is a polynomial term with a variable \(a\) raised to the power of 2. It represents the quadratic term.
• \(-2a\) - This is another polynomial term with the same variable \(a\), but raised to the power of 1, indicating the linear term.
• The equation lacks a constant term (or 0-power term) because there isn't a standalone number, which would typically be represented as \(c\) or just a number without \(a\).

Each polynomial term plays a specific role, defining the shape and properties of the equation’s graph.

Solving Algebraic Equations

Solving algebraic equations, especially quadratic ones, involves finding the values of the variable that make the equation true. The given problem \(8a^2 - 2a = 0\) is a good example for practicing solution strategies.

To solve the equation, it can be helpful to apply factoring techniques or the quadratic formula. Here, factoring can be effective:

• First, factor the equation. Both terms, \(8a^2\) and \(-2a\), contain \(a\), allowing you to factor out \(a\): \(a(8a - 2) = 0\).
• Next, using the principle of zero products, set each factor equal to zero: \(a = 0\) or \(8a - 2 = 0\).
• Solve the linear equation \(8a - 2 = 0\) for \(a\), which results in \(a = \frac{1}{4}\).

Hence, the solutions are \(a = 0\) and \(a = \frac{1}{4}\). This shows us that there are specific values where this equation holds true, showcasing how solving techniques can provide meaningful insights into the equation's nature.