Problem 76 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The discriminant of the given quadratic equation is -44, which indicates that there are two distinct complex solutions.

Step 1: Identify coefficients

Firstly, identify the coefficients a, b and c in the given quadratic equation. Here, \(a = 4\), \(b = -2\) and \(c = 3\).

Step 2: Calculate discriminant (D)

Substitute the values of a, b and c into the discriminant formula (\(b^{2} - 4ac\)). This results in \(-2^{2} - 4 * 4 * 3 = 4 - 48 = -44\). Therefore, the discriminant D is -44.

Step 3: Determine the type and number of solutions

Since the value of D is less than 0, the quadratic equation will have two distinct complex solutions. This is because a negative discriminant indicates that the roots are complex and different.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is an important concept in algebra. It is called 'quadratic' because it involves the squaring of a variable, typically written in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are known as coefficients, and \(a\) should not be zero since this would make the equation linear rather than quadratic.
Quadratic equations model many types of real-world problems, such as areas or projectile motions. Solving them helps us find the values of \(x\) that satisfy the equation. Depending on the discriminant, which we'll explore soon, the solutions can be real or complex. Quadratics are solved using different methods like factoring, completing the square, or using the quadratic formula. Among these, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is a universal method that provides solutions for any quadratic equation.

Exploring Complex Solutions

When dealing with quadratic equations, the discriminant helps us predict the nature of the roots. It is denoted by \(D\) and is calculated using the formula \(b^2 - 4ac\).
There are a few scenarios:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has exactly one real solution, also known as a repeated root.
• If \(D < 0\), the equation does not have real solutions; instead, it has two distinct complex solutions.

Complex solutions arise when \(D\) is negative. In such cases, the roots are expressed in terms of imaginary numbers. An imaginary number is expressed in the form of \(a + bi\), where \(i\) is the imaginary unit and equals \(\sqrt{-1}\). Complex solutions come in conjugate pairs, such as \(a + bi\) and \(a - bi\), reflecting the symmetry in their nature. Understanding complex solutions is crucial in advancing in higher mathematics.

The Importance of Coefficient Identification

Being able to correctly identify the coefficients in a quadratic equation is an essential first step in solving it. Each quadratic equation can be represented in the standard form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are known as the coefficients:

• \(a\): the coefficient of \(x^2\), which determines the "width" and direction of the parabola. It needs to be non-zero.
• \(b\): the coefficient of \(x\), affecting the position of the vertex along the horizontal axis.
• \(c\): the constant term, affecting the vertical position of the parabola on the graph, primarily the y-intercept.

In our example equation \(4x^2 - 2x + 3 = 0\), we identified \(a = 4\), \(b = -2\), and \(c = 3\). Identifying these coefficients accurately allows us to make further calculations, such as finding the discriminant or applying the quadratic formula effectively. Correct identification of coefficients is crucial because it forms the foundation for solving the quadratic equation and interpreting its graph or solutions correctly.