Problem 80 - College Algebra

Question

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$3 x^{2}-2 x=1$$

Step-by-Step Solution

Answer

The given equation has a discriminant of 16 and therefore, it has two distinct real solutions.

Step 1: Identify a, b, and c

First, rewrite the given equation in standard form \( ax^2 + bx + c = 0 \). The equation becomes \( 3x^2 - 2x - 1 = 0 \) where \( a = 3 \), \( b = -2 \), and \( c = -1 \)

Step 2: Compute the discriminant

Next, substitute values of a, b, and c into the formula for the discriminant. Therefore, \( D = (-2)^2 - 4*3*(-1) = 4 + 12 = 16 \)

Step 3: Determine the number and type of solutions

Because the discriminant is greater than 0, the equation has two distinct real solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra. They are polynomial equations of the second degree, generally written in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. This type of equation gets its name from the Latin word "quadratus," which means square, reflecting the squared term \( x^2 \) present in every quadratic equation.

Quadratic equations arise in various applications such as physics, engineering, and even in finance. Solving them involves finding the values of \( x \) that satisfy the equation, known as the roots or solutions. Whereas linear equations have a single solution, quadratic equations can have up to two solutions. The exact number and nature of these solutions depend on various factors, such as the values of \( a \), \( b \), and \( c \), as well as the associated discriminant value, which we'll address further in this article.

Real Solutions

Real solutions in the context of quadratic equations refer to the possible roots that are real numbers. These solutions arise depending on the value of the discriminant, \( D \), calculated as \( D = b^2 - 4ac \). This discriminant determines the nature and number of solutions an equation will have.

Here's how the discriminant affects the solutions:

• If \( D > 0 \), the equation has two distinct real solutions. This occurs when the curve of the quadratic, usually a parabola, intersects the \( x \)-axis at two different points.
• If \( D = 0 \), there is exactly one real solution. In this case, the parabola touches the \( x \)-axis exactly at one point, known as a repeated or double root.
• If \( D < 0 \), the equation has no real solutions. Instead, the solutions are complex, meaning they include imaginary numbers, and the parabola does not cross the \( x \)-axis.

Understanding discriminants helps predict and identify the nature of solutions without solving the equation directly, offering a quick insight into the behavior of quadratic equations.

Algebra

Algebra is like the language of mathematics. It helps organize numbers and symbols into expressions and equations—such as the quadratic equations we've been exploring. In algebra, rules and formulas are employed to solve for unknowns, often represented by letters like \( x \) or \( y \). This branch of mathematics builds upon basic arithmetic but allows for more complex problem-solving scenarios.

Quadratic equations are just one area within algebra that shows how this mathematical language operates. When solving them, you'll frequently deal with expressions, factors, and the manipulation of terms to isolate variables. The quadratic formula itself—\( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)—is a key algebraic tool, providing a direct method to find the solutions of any quadratic equation.

• Algebra enhances logical thinking by providing the framework to understand and manipulate equations.
• It also serves as a building block for advanced topics in mathematics.
• Algebra's techniques are useful not only in mathematics but also in sciences and engineering, where modeling and calculations are essential.

Mastery of algebra, including understanding concepts like quadratic equations and real solutions, is essential for anyone looking to work within STEM fields.