Problem 65 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the discriminant to determine the number of real solutions of the equation. $$ 3 y^{2}-4 y+5=0 $$

Step-by-Step Solution

Answer

The given quadratic equation is \(3y^2 - 4y + 5 = 0\). After calculating the discriminant \(D = (-4)^2 - 4(3)(5) = -44\), we see that \(D < 0\). Therefore, there are no real solutions for the given quadratic equation.

Step 1: Identify the coefficients

The quadratic equation is $$ 3y^2 - 4y + 5 = 0 $$ The coefficients are: - a = 3 - b = -4 - c = 5

Step 2: Compute the discriminant

Using the formula for the discriminant, $$ D = b^2 - 4ac $$ we substitute the values of a, b, and c: $$ D = (-4)^2 - 4(3)(5) $$ Calculate the result: $$ D = 16 - 60 $$ $$ D = -44 $$

Step 3: Determine the number of real solutions

Based on the value of the discriminant (D), the number of real solutions can be determined as follows: - If $$D > 0$$, there are two real solutions. - If $$D = 0$$, there is one real solution. - If $$D < 0$$, there are no real solutions. Since our discriminant $$D = -44$$, which is less than 0, there are: - No real solutions for the given quadratic equation.

Key Concepts

Quadratic Equations

Quadratic equations form one of the most fundamental concepts in algebra, and they appear frequently in various mathematical contexts. A quadratic equation is typically expressed in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) represent known numbers, with \( a \) not equal to zero, and \( x \) represents an unknown variable. The highest power of the unknown in these equations is 2, which leads to the term 'quadratic', from the Latin word 'quadratus' for 'square'.

The solutions to a quadratic equation are the values of \( x \) that make the equation true, and they are often referred to as the 'roots' of the equation. Quadratic equations can be solved through several methods, including factoring, completing the square, using the quadratic formula, or graphing. Understanding how to solve quadratic equations is critical not only in algebra classes but also in science, engineering, finance, and other quantitative fields.

Discriminant Calculation

The discriminant is a powerful tool in understanding the nature of the solutions to a quadratic equation without actually solving it. It is calculated using the formula \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the same coefficients from the quadratic equation in standard form. The discriminant reveals critical information about the roots of a quadratic equation.

Interpreting the Discriminant

The value of the discriminant has three possible outcomes concerning the real solutions of a quadratic equation:

• If \( D > 0 \), the quadratic equation has two distinct real solutions.
• If \( D = 0 \), the quadratic equation has exactly one real solution (also known as a repeated or double root).
• If \( D < 0 \), the quadratic equation has no real solutions; the solutions are complex numbers with a non-zero imaginary part.

The discriminant provides a quick way to ascertain whether the quadratic equation can be factored over the real numbers and whether the roots are real or complex.

Real Solutions of Equations

When we refer to 'real solutions' in the context of equations, we are talking about solutions that have values which can be plotted on the real number line. They are distinct from 'imaginary' or 'complex' solutions, which include an imaginary unit represented by \( i \), where \( i^2 = -1 \).

For a quadratic equation with real coefficients, real solutions are obtained when the discriminant is greater than or equal to zero. The solutions can be visualized on the graph of the quadratic equation as the points where the parabola intersects the x-axis. These solutions can be rational or irrational numbers, and they are of particular interest because they often represent real-world quantities that can be measured or observed.

Understanding whether an equation has real solutions is important in the context of both pure and applied mathematics, as it can influence how the equation is used to model real situations. For example, in physics, real solutions might represent feasible outcomes or states in a physical system.