Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Write the quadratic equation in general form. $$ 4 x^{2}-2 x=9 $$

Step-by-Step Solution

Answer

The quadratic equation in general form is \(4x^2 - 2x - 9 = 0\).

Step 1: Identification

Identify the given equation, which is \(4x^2 - 2x = 9\). Also, recognize that the general form of a quadratic equation is \(ax^2 + bx + c = 0\).

Step 2: Rearrange the Equation

Rearrange the given equation to the standard form by adding or subtracting terms on both sides. Here, subtract 9 on both sides to get: \(4x^2 - 2x - 9 = 0\).

Step 3: Reconfirm the General form

Confirm now that it matches the general form. The value of 'a' is 4, 'b' is -2 and 'c' is -9 in the equation.

Key Concepts

Algebraic Equations

Algebraic equations are the foundation of algebra. They are mathematical statements that assert the equality of two expressions and commonly involve one or more variables. For example, in the equation 4x^2 - 2x = 9, the variable x represents an unknown quantity that we are trying to determine.

Simplifying algebraic equations can involve several operations including addition, subtraction, multiplication, division, and factoring. The goal is to isolate the variable and solve for its value. In educational contexts, mastering algebraic equations is essential because they are used to model real-world problems and are the building blocks for more advanced topics in mathematics, science, and engineering.

Quadratic Equations

Quadratic equations are a specific type of algebraic equation where the highest power of the unknown variable is two. They can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero.

The equation from the exercise, 4x^2 - 2x - 9 = 0, is a quadratic equation with \(a=4\), \(b=-2\), and \(c=-9\). These equations are known for their characteristic parabolic graphs and their solutions can be found using various methods, such as factoring, completing the square, the quadratic formula, or graphically. Each solution method provides the same roots (solutions), which are the points where the parabola intersects the x-axis.

Rearranging Equations

Rearranging equations is a skill that involves the manipulation of expressions to form a desired structure without changing the inherent relationship between the variables. The process usually aims to isolate one variable, making the equation easier to analyze or solve.

When rearranging the given quadratic equation to match the general form \(ax^2 + bx + c = 0\), subtraction or addition is performed on both sides to preserve the equality. For instance, the original exercise illustrates this by subtracting 9 from both sides, leading to 4x^2 - 2x - 9 = 0. Understanding how to efficiently rearrange terms is crucial across mathematical disciplines and helps in solving equations accurately.