Problem 82 - College Algebra Essentials

Question

In Exercises \(75-82,\) compute the discriminant. Then determine the number and type of solutions for the given equation. $$ 3 x^{2}+4 x-2=0 $$

Step-by-Step Solution

Answer

The discriminant is 40. There are two distinct real solutions for the equation.

Step 1: Find the discriminant

Substitute the values of \(a\), \(b\) and \(c\) into the discriminant formula \(D = b^2 - 4ac\). Thus, the discriminant \(D = 4^2 - 4 * 3 * (-2) = 16 + 24 = 40\).

Step 2: Determine the nature of the roots

Establish the nature of the roots using the discriminant \(D\). If \(D > 0\), the equation has two distinct real roots. If \(D = 0\), there is one real root. If \(D < 0\), there are two complex roots. Here, because the discriminant \(D = 40\) is greater than zero, the equation has two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with a non-zero coefficient for the term in x². It has the standard form of ax2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The values of x that satisfy the equation are known as the 'roots' or 'solutions'.

A well-known method for solving quadratic equations is by using the quadratic formula, x = (-b ± √(b2 - 4ac)) / (2a), derived from the process of completing the square. The term under the square root, b2 - 4ac, is called the discriminant and plays a crucial role in determining the nature and number of roots the quadratic equation has.

Quadratic equations are deeply rooted in various areas of mathematics and sciences. They are used to model parabolic motion in physics, describe certain curves in geometry, and appear in optimization problems in algebra.

Nature of Roots

The discriminant in algebra is the key to understanding the nature of the roots of a quadratic equation. It is represented by a single symbol, usually 'D' or the Greek letter Delta (Δ), and is calculated using the formula D = b2 - 4ac. The value of the discriminant determines the type and number of roots:

• If D > 0, the quadratic equation has two distinct real roots.
• If D = 0, there is exactly one real root, also known as a repeated or a double root.
• If D < 0, there are two complex roots, which are conjugates of each other.

The discriminant reveals whether the parabola described by the quadratic equation intersects the x-axis (real roots) or not (complex roots). This concept is particularly significant as it helps us predict the behavior of a quadratic function without necessarily computing the exact roots.

Solving Quadratic Equations

Solving quadratic equations involves finding values of the variable that make the equation true. There are several methods to solve such equations:

• Factoring: Writing the quadratic as the product of two binomials, if possible.
• Using the square roots: Applicable when the quadratic equation can be written in a form such that one side is a perfect square.
• Completing the square: Transforming the quadratic equation to have a perfect square on one side.
• Quadratic formula: This is the most comprehensive method and works for all quadratic equations. As mentioned earlier, the solutions can be found using x = (-b ± √(b2 - 4ac)) / (2a).

It is important for students to practice these methods and understand the conditions under which each method is most efficient. For instance, the factoring method is quickest when the quadratic is easily factorable, while the quadratic formula is the go-to method for more complex equations. Highly effectively grasping these concepts ensures that students are able to approach quadratic equations with a robust toolkit, ready to tackle various problems.