Problem 80 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The discriminant is \(-8\), thus the given quadratic equation has two distinct complex solutions.

Step 1: Write the equation in the standard form

To write the equation \(3x^2 = 2x - 1\) in the standard form \(ax^2 + bx + c = 0\), subtract \(2x\) and add \(1\) to both sides of the equation. This results in \(3x^2 - 2x + 1 = 0\).

Step 2: Identify values of \(a\), \(b\), and \(c\)

In the equation \(3x^2 - 2x + 1 = 0\), here \(a = 3\), \(b = -2\), and \(c = 1\). These coefficients are used in the discriminant formula.

Step 3: Substitute \(a\), \(b\), and \(c\) into the discriminant formula

The formula for the discriminant is \(b^2 - 4ac\). Substituting the values for \(a\), \(b\), and \(c\) from Step 2, we get \((-2)^2 - 4 * 3 * 1 = 4 - 12 = -8\).

Step 4: Determine the nature of the roots

The discriminant is \(-8\), which is less than zero. Therefore, the given quadratic equation has two distinct complex solutions.

Key Concepts

Nature of Roots

The nature of roots of a quadratic equation is determined by the value of its discriminant. The discriminant is found using the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in its standard form. The value of the discriminant reveals important information about the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real double root, sometimes called a repeated root.
• If the discriminant is negative, like in our example, the equation has two distinct complex roots.

In this exercise, we found the discriminant to be \(-8\), indicating that the quadratic equation yields two complex solutions rather than real numbers.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the variable or unknown. Arranging a quadratic equation into this form is essential before applying methods to solve it, such as the quadratic formula or calculating the discriminant.

For instance, the quadratic equation we dealt with started as \(3x^2 = 2x - 1\). To convert this into standard form, we rearranged the terms to become \(3x^2 - 2x + 1 = 0\). This transformation makes it easier to identify the coefficients and work with the equation systematically.

Complex Solutions

Complex solutions arise in quadratic equations when the discriminant is negative. In this scenario, real numbers aren't sufficient to express the solutions, which is where complex numbers come into play. Complex numbers are numbers that have a real part and an imaginary part, denoted usually as \(a + bi\), where \(i\) stands for the imaginary unit \(\sqrt{-1}\).

In the example given, the discriminant of the quadratic equation \(3x^2 - 2x + 1 = 0\) was negative. Therefore, it has two complex solutions. These solutions are conjugates of each other, meaning if one solution is \(a + bi\), the other is \(a - bi\). Understanding the concept of complex solutions helps in working with a wider range of quadratic equations beyond those with real number roots.

Quadratic Equation Coefficients

In the standard form of a quadratic equation \(ax^2 + bx + c = 0\), the values of \(a\), \(b\), and \(c\) are called the coefficients. They play a crucial role in determining the properties and solutions of the equation. Here's how each coefficient contributes:

• \(a\), the leading coefficient, affects the direction and width of the parabola represented by the equation. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• \(b\), the linear coefficient, influences the axis of symmetry of the parabola and, along with \(a\), affects the vertex's horizontal position.
• \(c\), the constant term, controls the vertical position of the parabola relative to the y-axis.

In our example, the coefficients are \(a = 3\), \(b = -2\), and \(c = 1\). Identifying these coefficients allows us to substitute them into formulas like the discriminant to further analyze the nature of the equation’s solutions.