Problem 13 - Algebra 2

Question

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(2 x^{2}-4 x+1=0\)

Step-by-Step Solution

Answer

The discriminant is 8; there are two distinct real roots.

Step 1: Identify coefficients

Start by identifying the coefficients in the quadratic equation \(2x^{2}-4x+1=0\). Here, \(a = 2\), \(b = -4\), and \(c = 1\).

Step 2: Calculate the discriminant

Use the formula for the discriminant \(D = b^2 - 4ac\). Substitute the values to find \(D = (-4)^2 - 4(2)(1) = 16 - 8 = 8\).

Step 3: Determine the roots based on the discriminant

Since the discriminant \(D = 8\) is positive, the quadratic equation has two distinct real roots.

Key Concepts

Discriminant

In quadratic equations, understanding the discriminant is key. The discriminant is part of the quadratic formula, which helps us solve for the roots of the equation. It is calculated using the formula:

• \[D = b^2 - 4ac\]

Here, the discriminant "\(D\)" uses the coefficients of the quadratic equation: "\(a\), \(b\), and \(c\)" from the general form of the quadratic equation \(ax^2 + bx + c = 0\). The value of the discriminant tells us about the nature of the roots. It informs us whether the roots of the quadratic equation are real or complex. A positive discriminant indicates that the roots are real and distinct, whereas a discriminant of zero shows that the roots are real and equal. A negative discriminant means the roots are imaginary or complex. Understanding this value helps predict the behavior of the quadratic graph on the coordinate plane.

Real Roots

The concept of real roots is fundamental when solving quadratic equations. Real roots are the solutions to the quadratic equation, and they can be found using the quadratic formula or by factoring, when applicable. When we use the discriminant to determine the number of real roots, it affects how we view the solutions:

• A positive discriminant indicates two distinct real roots.
• A discriminant of zero points to exactly one real root, or a repeated root.
• A negative discriminant means no real roots exist, only complex roots.

Real roots can be visualized as the points where the graph of the quadratic equation intersects the x-axis. When dealing with real roots, the parabola that represents the quadratic function will cross (or just touch) the x-axis. This visual representation makes it easier to understand the solutions and how they relate to the equation itself.

Coefficients

In any quadratic equation, the coefficients \(a\), \(b\), and \(c\) hold significant roles. They are the constants in the equation \(ax^2 + bx + c = 0\). Each one affects the shape and position of the parabola when graphing the equation. Here’s how they impact the equation:

• \(a\) - It determines the direction of the parabola (upwards if positive, downwards if negative) and how "wide" or "narrow" it is. A larger \(a\) causes a steeper parabola.
• \(b\) - This coefficient affects the axis of symmetry of the parabola and the vertex's horizontal position. Changes in \(b\) will shift the parabola along the x-axis.
• \(c\) - It is the y-intercept of the graph. This point is where the parabola crosses the y-axis.

Understanding the impact of these coefficients is important for manipulating and interpreting quadratic functions. When solving for roots using the quadratic formula, these coefficients are crucial, as they directly influence the values of the discriminant and ultimately, the nature of the roots.