Problem 6 - Basic Technical Mathematics with Calculus

Question

Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify \(a, b,\) and \(c,\) with \(a>0 .\) Otherwise, explain why the resulting form is not quadratic. $$x\left(2 x^{2}+5\right)=7+2 x^{2}$$

Step-by-Step Solution

Answer

The equation is not quadratic; it is cubic (degree 3).

Step 1: Expand and Simplify Equation

First, distribute the \(x\) across \(2x^2+5\) on the left side of the equation: \[ x \times (2x^2 + 5) = 2x^3 + 5x \]. The equation becomes: \[ 2x^3 + 5x = 7 + 2x^2 \]. Now, let's rewrite everything on one side: \[ 2x^3 + 5x - 7 - 2x^2 = 0 \]. Rearrange in decreasing powers of \(x\) to make it: \[ 2x^3 - 2x^2 + 5x - 7 = 0 \].

Step 2: Identify Quadratic Characteristics

A quadratic equation takes the form \(ax^2 + bx + c = 0\). In this equation, the highest degree term is \(2x^3\) which indicates this is a cubic equation, not a quadratic equation. Since the highest degree of \(x\) is 3, this equation doesn't fit the quadratic form \(ax^2 + bx + c = 0\). Therefore, it cannot be defined by \(a\), \(b\), and \(c\) in terms of quadratic equations.

Key Concepts

Cubic Equations

Cubic equations are a type of polynomial equation where the highest degree of the variable, often denoted as "x," is three. In simpler terms, the equation includes a term that has the variable raised to the third power, like \(x^3\). These equations can represent more complex relationships than quadratic equations, which only include terms up to \(x^2\).

• They usually take the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).
• The inclusion of an \(x^3\) term means they will have at least one real root, though they can also have complex roots.
• Like quadratics, solving cubic equations often requires techniques such as factoring, the Rational Root Theorem, or numerical methods for finding approximate roots.

Understanding cubic equations is important as they frequently describe real-world phenomena such as volume in three dimensions, kinetic energy in physics, and certain economic models.

Polynomial Degree

The degree of a polynomial is determined by the highest power of the variable present in the equation. This concept is crucial in distinguishing between different types of polynomial equations, such as linear, quadratic, cubic, and beyond.

• Polynomial degree indicates the number of roots or solutions the polynomial can have, though not all roots may be real.
• For example, a quadratic polynomial like \(ax^2 + bx + c = 0\) has a degree of 2, indicating it can have up to two roots.
• In contrast, a cubic polynomial such as \(ax^3 + bx^2 + cx + d = 0\) has a degree of 3, potentially possessing three roots.

Understanding polynomial degree aids in graphing these equations and predicting the general shape and behavior of the graph based on the leading term.

Identifying Quadratic Forms

Identifying whether an equation is quadratic involves recognizing its structure and comparing it to the standard quadratic form. A quadratic equation typically takes the format \(ax^2 + bx + c = 0\), where \(a\) is a non-zero coefficient.

• The key features of a quadratic equation include the presence of an \(x^2\) term, which should be the highest degree term, and one or more \(x\) or constant terms.
• If an equation contains terms with an \(x^3\) or higher power, it is not quadratic.
• When assessing whether an equation is quadratic, simplify and rearrange the terms to check if it fits the standard quadratic form.

Recognizing quadratic forms is essential in solving these equations and using methods like the quadratic formula, factoring, or graphing to find solutions.