Problem 70 - College Algebra

Question

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(3 x^{2}=2 x-1\)

Step-by-Step Solution

Answer

The discriminant of the equation \(3x^2 - 2x + 1 = 0\) is -8. Since the discriminant is less than zero, the equation has two complex roots.

Step 1: Identify the coefficients a, b, c

From the given equation \(3x^2 = 2x - 1\), we need to rewrite it to the standard quadratic equation form \(ax^2 + bx + c = 0\). So, our equation becomes \(3x^2 - 2x + 1 = 0\). From this, identify the coefficients a, b and c. Here, \(a = 3\), \(b = -2\), and \(c = 1\).

Step 2: Calculate the discriminant

Substitute the values of a, b, and c into the discriminant formula \(b^2 - 4ac\). Hence, Discriminant \(D = (-2)^2 - 4*3*1 = 4 - 12 = -8\).

Step 3: Analyze discriminant to determine number and type of solutions

Having calculated the discriminant as -8 which is less than 0, it indicates that the quadratic has two complex roots.

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression that equates to zero, usually written in the form \( ax^2 + bx + c = 0 \). This equation is characterized by three important elements: the coefficients \(a\), \(b\), and \(c\). In this type of equation:

• \(a\) is the coefficient of \(x^2\), which ensures the equation is quadratic and not of a lower order (like linear).
• \(b\) is the coefficient of \(x\), affecting the symmetry and position of the parabola on a graph.
• \(c\) is the constant term, shifting the parabola up or down on the graph.

To solve a quadratic equation, one can employ various methods such as factoring, completing the square, or using the quadratic formula. Each method aims to find the value(s) of \(x\) that satisfy the equation. Indeed, the quadratic formula can directly tell us about the solutions using the discriminant, which is crucial in understanding the nature of the roots.

Complex Roots

When solving a quadratic equation, the discriminant \(b^2 - 4ac\) plays a vital role in determining the nature of the roots. Complex roots occur when the discriminant is less than zero. This means that the equation does not have real number solutions, but instead, the roots are complex numbers. A complex number typically has a real part and an imaginary part and is of the form \(a + bi\), where \(i\) is the imaginary unit \(\sqrt{-1}\).The presence of complex roots shows that the quadratic equation's graph does not intersect the x-axis, as real roots would. Instead:

• There are two complex roots, which are conjugates of each other (e.g., \(2 + 3i\) and \(2 - 3i\)).
• These roots add another layer of diversity in mathematical solutions, expanding beyond the real number line.
• Solving quadratics with complex roots involves algebraic techniques and understanding of imaginary numbers.

By understanding these complex solutions, students gain insight into more sophisticated mathematical concepts that extend beyond typical numbers encountered in basic mathematics.

Coefficients

Coefficients are numerical or constant factors in terms of an expression or equation. In the context of a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients \(a\), \(b\), and \(c\) assume specific roles:

• \(a\) is the coefficient of the quadratic term \(x^2\). It defines the opening direction and the width of the parabola. A positive \(a\) opens upwards, while a negative \(a\) opens downwards.If \(a\) is large, the parabola is narrow; if small, it is wide.
• \(b\) is the coefficient of the linear term \(x\). It influences how the parabola shifts horizontally in relation to the y-axis on the graph.
• \(c\) is the constant or the y-intercept of the parabola, showing the point where the parabola meets the y-axis.

Understanding coefficients is crucial for graphing parabolas and predicting the behavior of quadratic relationships. By analyzing these coefficients, one can predict the number and nature of roots through the discriminant or visualize transformational shifts in the graph.