Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 3 x^{2}-5 x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{5 + \sqrt{13}}{6} \) and \( x = \frac{5 - \sqrt{13}}{6} \).

Step 1: Identify Coefficients

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Identify the coefficients: \( a = 3 \), \( b = -5 \), and \( c = 1 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) is calculated using the formula \( \Delta = b^2 - 4ac \). Plug in the values: \( \Delta = (-5)^2 - 4 \times 3 \times 1 = 25 - 12 = 13 \).

Step 3: Determine Nature of Roots

Since the discriminant \( \Delta = 13 \) is greater than zero, it implies the equation has two distinct real solutions.

Step 4: Apply the Quadratic Formula

Substitute \( a = 3 \), \( b = -5 \), and \( c = 1 \) into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{13}}{2 \times 3} \]Simplify the expression to find the values of \( x \).

Step 5: Simplify the Solution

Solve for \( x \) using the expression obtained:\[ x = \frac{5 \pm \sqrt{13}}{6} \]Thus, the solutions are \( x = \frac{5 + \sqrt{13}}{6} \) and \( x = \frac{5 - \sqrt{13}}{6} \).

Key Concepts

Understanding the Discriminant

In any quadratic equation, the discriminant plays a pivotal role in understanding the nature of the roots of the equation. The discriminant, denoted as \( \Delta \), is part of the quadratic formula which is \( b^2 - 4ac \). We'll find this by plugging in our coefficients, which in our example are \( a = 3 \), \( b = -5 \), and \( c = 1 \).

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution.
• If \( \Delta < 0 \), no real solutions exist, meaning the roots are complex or imaginary.

In the given problem, you saw \( \Delta = 13 \), a positive number which indicates the existence of two distinct real solutions.

Finding Real Solutions

Real solutions in the context of quadratic equations refer to values of \( x \) that satisfy the equation where the roots are non-complex. These solutions are found using the discriminant and the quadratic formula. If the discriminant is greater than zero, like \( 13 \) in our example, we immediately know that our equation will have real solutions.

The actual step to find these solutions involves substituting the values of \( a \), \( b \), and \( c \) into the quadratic formula and solving for \( x \). For example, solving \( 3x^2 - 5x + 1 = 0 \) results in two solutions as shown earlier. This provides a clear path to find all possible real solutions effectively.

The Quadratic Formula

The quadratic formula is a robust tool used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Using it involves straightforward steps:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Substitute these values into the quadratic formula.
• Simplify the resulting expression to find the values of \( x \).

This method is particularly powerful because it provides a way to find real, imaginary, or repeated solutions for any quadratic equation. It's an essential tool across multiple domains of mathematics and science.

Quadratic Equations in Mathematics Education

Quadratic equations play a crucial role in mathematics education due to their prominence in various fields and their fundamental nature. By understanding quadratic equations, students gain exposure to key mathematical concepts like equations, roots, and functions.

Here are a few reasons why it's important:

• They form the basis for growth in algebra and help in understanding more complex mathematical problems.
• They illustrate important concepts such as symmetry and function transformations.
• Real-world applications include physics formulas, financial models, and engineering calculations.

Learning about quadratic equations, discriminants, and solutions prepares students for numerous STEM careers and enhances problem-solving skills, making mathematics a more approachable and practical subject.