Problem 16 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 2 x^{2}+5 x=0 $$

Step-by-Step Solution

Answer

Question: Identify the values of a, b, and c in the given quadratic equation: \(2x^2 + 5x = 0\) Answer: In the given quadratic equation, the values are: \(a = 2, \: b = 5, \: c = 0\).

Step 1: Identify the coefficient 'a'

The coefficient 'a' is the number before the \(x^2\) term. In the given equation, it is 2. Thus, \(a = 2\).

Step 2: Identify the coefficient 'b'

The coefficient 'b' is the number before the 'x' term. In the given equation, it is 5. Thus, \(b = 5\).

Step 3: Identify the coefficient 'c'

The constant term 'c' is the term without x. In the given equation, there is no constant term, which means it is 0. Thus, \(c = 0\). The values of a, b, and c in the given quadratic equation are: $$ a = 2, \quad b = 5, \quad c = 0 $$

Key Concepts

Coefficients

Coefficients are vital components in quadratic equations, offering insight into the behavior and solutions of these expressions. They are the numerical factors that multiply with the variables. In a standard quadratic equation, written as \(ax^2 + bx + c = 0\), the coefficients are the numbers \(a\), \(b\), and \(c\). Each of these numbers holds a specific role:

• The quadratic coefficient (\(a\)) is associated with the \(x^2\) term.
• The linear coefficient (\(b\)) is linked to the \(x\) term.
• The constant term (\(c\)) stands on its own without an \(x\) attached to it.

Understanding these coefficients is crucial in determining the shape and position of a parabola on a graph. Let's explore each one in detail.

Quadratic Coefficient

The quadratic coefficient is represented by the letter \(a\) in a quadratic equation. It is the key factor that affects the curvature or concavity of the parabola described by the quadratic equation.

• If \(a > 0\), the parabola opens upwards, resembling a smile.
• If \(a < 0\), the parabola opens downwards, similar to a frown.
• The value of \(a\) also influences the width of the parabola. Larger values result in a narrower parabola, while smaller values produce a wider one.

In the example equation \(2x^2 + 5x = 0\), the quadratic coefficient \(a\) is 2, indicating an upward-opening parabola with a relatively standard width.

Linear Coefficient

The linear coefficient, noted as \(b\) in the quadratic equation, directly affects the slope and position of the parabola on the graph. It essentially tilts the parabola either left or right. While \(b\) plays less of a role in the shape compared to \(a\), it is crucial in determining the parabola's intersection points with the y-axis.

• Changes in \(b\) adjust where the vertex of the parabola is horizontally aligned.
• A higher absolute value of \(b\) may cause the parabola to shift more aggressively across the axis.

In the equation \(2x^2 + 5x = 0\), \(b\) is 5, contributing to the angle at which the parabola intersects the y-axis.

Constant Term

The constant term, symbolized by \(c\), is the number in the equation that stands alone without being multiplied by a variable. It sets the vertical position of the entire parabola on the graph, indicating where the parabola will cross the y-axis. If \(c\) were to change:

• A positive \(c\) would lift the parabola upwards on the graph.
• A negative \(c\) would push the parabola downwards.
• \(c = 0\) means the parabola passes through the origin (0,0), as is the case in \(2x^2 + 5x = 0\).

In our specific example, \(c\) is 0, showing that the parabola intersects exactly at the origin, aligning it symmetrically on the graph.