Problem 13 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 7 a^{2}+a-8=0 $$

Step-by-Step Solution

Answer

Answer: In the given quadratic equation, the values are a = 7, b = 1, and c = -8.

Step 1: Compare the given equation to the general form of a quadratic equation

Compare the given equation, $$ 7a^2 + a - 8=0 $$ to the general form of a quadratic equation: $$ ax^2 + bx + c = 0 $$

Step 2: Identify the values of a, b, and c

In the given equation, we can see that the coefficients of the corresponding terms are: a: 7 (the coefficient of the \(a^2\) term) b: 1 (the coefficient of the a term) c: -8 (the constant term)

Step 3: Write down the values of a, b, and c

Now that we have identified the values of a, b, and c, we can write them down: a = 7 b = 1 c = -8

Key Concepts

Coefficients

Coefficients play a critical role in quadratic equations. In essence, they are the numerical factors that multiply the variables in a polynomial equation. When dealing with quadratic equations, the coefficients define the unique features and shape of the parabola represented by the equation. For instance, let's consider the equation:
\(7a^2 + a - 8 = 0\).
In this quadratic equation:

• The number 7 is the coefficient of the squared term \(a^2\).
• The number 1 is the coefficient of the linear term \(a\).
• Lastly, -8 is a constant term, which sometimes is also referred to as a coefficient for simplicity.

Understanding coefficients is crucial for solving, graphing, or analyzing quadratic equations, as they directly influence the roots and the vertex of the parabola.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is a fundamental aspect when dealing with quadratic expressions. It is formally expressed as:
\(ax^2 + bx + c = 0\),
where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). This form is essential because it provides a straightforward way to apply various mathematical methods to analyze and solve the equation.

• The term \(ax^2\) represents the quadratic component where "\(a\)" is the coefficient of \(x^2\).
• The term \(bx\) represents the linear component with "\(b\)" as its coefficient.
• "c" is the constant term, without a variable attached.

This standard form is the first step in many procedures involving quadratic equations, like factoring, using the quadratic formula, or completing the square. It serves as a foundational template to ensure that a problem can be approached systematically.

Identifying Terms in Polynomials

When analyzing polynomials, recognizing and differentiating between the terms is crucial. A polynomial is an expression composed of variables and coefficients, structured as the sum of terms. Each term consists of a coefficient and a variable raised to a power. In a quadratic equation like:
\(7a^2 + a - 8 = 0\),
we can identify different kinds of terms:

• Quadratic Term: This is the term with the highest power of the variable. Here it's \(7a^2\), where "7" is the coefficient, and the variable "\(a\)" is raised to the second power.
• Linear Term: This term involves the variable to the first power. In this example, the linear term is "\(a\)" or equivalently "1\times a" where "1" is the coefficient.
• Constant Term: Without any variable, it's a standalone number. Here, it is "-8."

Being able to pick out each term accurately is a skill that simplifies the process of solving polynomial equations. It helps in organizing and applying the correct algebraic principles when tackling more complex algebraic structures.