Problem 2 - Elementary Algebra

Question

Classify the following equations in terms of their degree. $$ 9 x^{2}+5 x-6=3 $$

Step-by-Step Solution

Answer

Answer: The degree of the given equation is 2, and it is classified as a quadratic equation.

Step 1: Identify the given equation

Begin by identifying the equation given in the exercise: $$ 9x^2+5x-6=3 $$

Step 2: Isolate the equation to one side

In order to simplify the equation, we will first isolate the equation to one side by moving the constant term to the other side of the equal sign: $$ 9x^2+5x-6-3=0 $$ $$ 9x^2+5x-9=0 $$

Step 3: Identify the highest exponent of the variable

Now, we will identify the highest exponent of the variable x in the equation: In the given equation, the highest exponent of x is 2, in the term \(9x^2\)

Step 4: Classify the equation based on its degree

Using the information obtained in Step 3, we can now classify the equation based on its degree. The equation has a degree of 2 because the highest exponent of the variable x is 2. The equation $$9x^2+5x-9=0$$ is a second-degree equation, also known as a quadratic equation.

Key Concepts

Degree of an Equation

When we talk about the degree of an equation in algebra, we refer to the highest power (exponent) of the variable present. It's a simple yet critical characteristic that helps to classify equations.
For instance, in the equation \[9x^2 + 5x - 6 = 3\],
the highest exponent of the variable x is 2, found in the term \(9x^2\). Thus, the degree of this equation is 2. Determining the degree is crucial because it dictates the behavior of the equation, including the number of solutions it may have and the shape of its graph. A linear equation has a degree of 1 and typically has a single solution, forming a straight line on a graph. On the other hand, equations with higher degrees, such as quadratic equations (degree 2), can have more complex solutions and graphs.

Algebraic Equations

Algebraic equations are the bread and butter of high school mathematics. They consist of numbers, variables, and operations that are set equal to each other. These equations can range from simple linear equations to more complex ones involving higher powers and multiple variables.
An equation is like a scale that needs to be balanced. The primary goal when working with algebraic equations is to isolate the variable, allowing us to find its value. In doing so, we follow a set of rules and operations—such as addition, subtraction, multiplication, and division—to maintain the balance of the equation.
In the context of our example \[9x^2 + 5x - 6 = 3\],
we first make sure that all terms involving the variable are on one side of the equation and the constant terms are on the other. This simplification is an important step to solve the equation and shows the standard form of equations we often seek in algebra.

Quadratic Equations

Quadratic equations are a specific type of algebraic equation that play a significant role in various areas of mathematics and applied sciences. They are second-degree equations, which means the highest exponent of the variable is 2.
A general form of a quadratic equation is \(ax^2 + bx + c = 0\),
where a, b, and c are constants, with 'a' being non-zero. The shape of the graph of any quadratic equation is a parabola, which can open upwards or downwards depending on the sign of 'a'.
They can be solved by various methods, including factoring, completing the square, using the quadratic formula, or graphically. The exercise we are looking at (\(9x^2 + 5x - 9 = 0\)) is a quintessential example of a quadratic equation. Identifying the equation as quadratic is critical because it tells us that we can expect up to two real solutions and that we will use specific methods designed for quadratic equations to solve it.