Problem 41 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possib $$\frac{3}{4} x^{2}-\frac{5}{2} x-2=0$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{3}{4}x^{2} - \frac{5}{2}x - 2 = 0\) is \(x =\frac{2 \pm \sqrt{13}}{3}\).

Step 1: Identify the Constants a, b, and c

In the equation \(\frac{3}{4}x^{2} - \frac{5}{2}x - 2 = 0\), the values of a, b, and c are as follows: a = \(\frac{3}{4}\), b = \(-\frac{5}{2}\), and c = -2.

Step 2: Substitute the Constants into the Quadratic Formula

Substituting these values into the quadratic formula gives \(x = \frac{-(-\frac{5}{2}) \pm \sqrt{(-\frac{5}{2})^{2} - 4 \times (\frac{3}{4}) \times -2}}{2 \times \frac{3}{4}}\).

Step 3: Simplify the Equation

Simplify the above equation to get \(x = \frac{5}{4} \pm \frac{\sqrt{(\frac{5}{2})^{2} + 6}}{3/2}\).

Step 4: Calculate the two possible values of x

Solving the above equation gives the two possible values of x, which are \(x = \frac{5}{4} + \frac{\sqrt{(\frac{5}{2})^{2} + 6}}{3/2}\) and \(x = \frac{5}{4} - \frac{\sqrt{(\frac{5}{2})^{2} + 6}}{3/2}\).

Step 5: Simplify the solutions

After simplifying these solutions, we get \( x =\frac{2 \pm \sqrt{13}}{3}\).

Key Concepts

quadratic formula

The quadratic formula is a powerful tool in algebra, used to solve quadratic equations. A quadratic equation is a second-degree polynomial equation that can be structured as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula helps find the values of \(x\) that satisfy the equation by using the special formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula might seem complex at first glance, but it systematically solves for \(x\) when you know \(a\), \(b\), and \(c\).

• Standard Form Conversion: Ensure your equation is in the form \(ax^2 + bx + c = 0\).
• Identify Constants: Isolate \(a\), \(b\), and \(c\).
• Plug and Play: Substitute these values into the quadratic formula.
• Discriminant Insight: The term \(b^2 - 4ac\), known as the discriminant, reveals the nature of the roots: if it's positive, there are two real solutions; if zero, one real solution; if negative, the solutions are complex.

Understanding these steps allows anyone to handle even the trickiest quadratic equations, making it an invaluable technique in algebra.

algebraic simplification

Algebraic simplification is the process of making an algebraic expression more manageable and easier to deal with. This often involves reducing fractions, combining like terms, and applying basic operations to simplify the equation's appearance. Let's break down some key steps:

• Combine Like Terms: Sum or subtract terms that have the same variables and exponents. This can significantly cut down the complexity of an expression.
• Factor and Simplify Expressions: Look for common factors or use methods like reverse distribution to make expressions more elegant.
• Rationalize Fractions: Eliminate fractions where possible, especially in the context of the quadratic formula, where fractions are a common outcome.

In our example, simplifying the equation involved reducing the expression inside the square root and handling the fraction in the formula. This resulted in the final simplified form \( x = \frac{2 \pm \sqrt{13}}{3} \), showcasing the value of careful simplification. When tackling algebraic problems, taking the time to simplify can save you a lot of work and make solutions clearer.

solving equations

Solving equations is a fundamental skill in algebra, encompassing various techniques to find the variable values that satisfy the given expression. The goal is to 'solve for \(x\)' or the required variable by following systematic steps:

• Rearrangement: Move all terms to one side to set everything to zero, if not already in standard form.
• Selection of Method: Choose the appropriate method (quadratic formula, factoring, or completing the square) based on the equation's structure.
• Validation: After solving, substituting back into the original equation ensures accuracy.

In our exercise, we opted for the quadratic formula, a reliable choice for any equation of the form \(ax^2 + bx + c = 0\). By first noting our equation's constants and directly substituting them into the formula, we land on our solutions. With practice, solving becomes more intuitive and less error-prone. Regardless of the method, always double-check results to confirm they meet all original equation conditions.