Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 1-10, write the quadratic equation in general form. $$ 2 x^{2}=3-5 x $$

Step-by-Step Solution

Answer

The quadratic equation in general form is \(2x^2 + 5x - 3 = 0\).

Step 1: Identify the quadratic equation

The given equation is \(2x^2 = 3 - 5x\).

Step 2: Rearrange the equation

To change the equation to the general form, we need to bring all the terms on one side of the equation by adding \(5x\) on both sides. So equation becomes \(2x^2 + 5x - 3 = 0\).

Step 3: Write the quadratic equation in general form

The quadratic equation in general form is \(2x^2 + 5x - 3 = 0\).

Key Concepts

General Form of Quadratic Equations

The general form of a quadratic equation is a classic way to represent quadratic relationships. It's written as \(ax^2 + bx + c = 0\). This formula is key because it helps in identifying the structure of any quadratic equation. Here, \(a\), \(b\), and \(c\) are constants:

• \(a\) is the coefficient of \(x^2\). It determines the width and the direction of the parabola when graphed.
• \(b\) is the coefficient of \(x\). It affects the parabola's position along the x-axis.
• \(c\) is the constant term. It tells us where the graph intersects the y-axis.

By having equations in this form, you can easily apply mathematical methods to find solutions or analyze the equation further. Simple as it seems, recognizing and transforming an equation into this form is a fundamental skill in algebra. Practicing it helps to understand more complex concepts that build upon quadratic relationships.

Rearranging Equations

Rearranging an equation is like re-organizing a room. All elements need to be considered and placed correctly. In our problem, we started with \(2x^2 = 3 - 5x\). To put it into general form, all terms must be on one side of the equation:

• We do this by performing operations on both sides. Just like moving furniture, the aim is to keep everything balanced.
• Adding \(5x\) to both sides of the equation ensures that all terms come together on the left side: \(2x^2 + 5x = 3\).
• Then, subtracting 3 from both sides finishes the arrangement: \(2x^2 + 5x - 3 = 0\).

Using steps like this ensures that the equation is properly prepared for solving. Properly rearranging equations lays the groundwork for solving them accurately. It’s about patience and careful, systematic work. Understanding how to rearrange equations is a critical step, as it prepares the equation for further analysis or solving.

Solving Quadratic Equations

Once a quadratic equation is in general form, it's ready to be solved. There are several methods for finding solutions:

• Factoring: Decomposition of the quadratic into two binomials if possible. It's efficient but not always applicable.
• Using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This is a universal method that works for any quadratic equation in general form.
• Completing the Square: Rewriting the equation to a perfect square trinomial. This is more advanced, often used to derive the quadratic formula itself.
• Graphical Method: Plot the equation and look for x-intercepts. Visualizing the solution's location on the graph gives an intuitive understanding.

Each method has its benefits and is chosen based on the specifics of the equation. For example, if an equation is easily factorable, factoring is quickest. The quadratic formula is very reliable, especially when the equation resists simple factoring. Understanding these methods equips you to tackle any quadratic equation you encounter. It's about selecting the right tool for the job, ensuring clarity and precision in your mathematical work.