Problem 14 - 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? \(x^{2}+3 x+8=5\)

Step-by-Step Solution

Answer

Discriminant: -3; Two complex conjugate roots.

Step 1: Simplify the Equation

To begin, we need to express the quadratic equation in the standard form, which is \( ax^2 + bx + c = 0 \). Start by subtracting 5 from both sides: \( x^2 + 3x + 8 - 5 = 0 \), simplifying to \( x^2 + 3x + 3 = 0 \).

Step 2: Identify Coefficients

Identify the coefficients from the simplified quadratic equation \( x^2 + 3x + 3 = 0 \). Here, \( a = 1 \), \( b = 3 \), and \( c = 3 \).

Step 3: Calculate the Discriminant

The discriminant \( D \) of a quadratic equation is given by \( D = b^2 - 4ac \). Substitute the identified values to get \( D = 3^2 - 4 \times 1 \times 3 \). Calculate to find \( D = 9 - 12 = -3 \).

Step 4: Determine the Roots Based on the Discriminant

The value of the discriminant determines the nature of the roots. If \( D > 0 \), there are two distinct real roots; if \( D = 0 \), there is one real root (a repeated root); if \( D < 0 \), there are two complex conjugate roots. Since \( D = -3 \), which is less than zero, the equation has two complex conjugate roots.

Key Concepts

Discriminant

In quadratic equations, the term "discriminant" plays a critical role in understanding the nature of the roots. The discriminant is calculated using the formula:\[ D = b^2 - 4ac \]Where:

• \(a\), \(b\), and \(c\) are coefficients of the quadratic equation in standard form \(ax^2 + bx + c = 0\).
• \(b^2\) is the square of the coefficient of \(x\).

The value of \(D\) informs us about the type of roots the quadratic equation has. A positive discriminant (\(D > 0\)) indicates two distinct real roots. If the discriminant equals zero (\(D = 0\)), the quadratic has exactly one real root, also known as a repeated or double root. Negative discriminants (\(D < 0\)) suggest that the equation has complex roots. Understanding this early on helps us determine how we should proceed in finding the roots.

Roots of a Quadratic Equation

The roots or solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These can be found using several methods:

• Factoring, if the equation is factorable.
• Completing the square, which helps in rewriting the equation into a perfect square.
• The quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Each of these methods requires an understanding of the discriminant, as it can guide you on which approach is more feasible. The roots can either be real and distinct, real and repeated, or complex, largely dictated by the discriminant's value. Evaluating the roots is crucial in many mathematical applications such as solving real-life problems related to physics, engineering, and economics.

Complex Roots

Complex roots arise in quadratic equations when the discriminant is negative. When \( D < 0 \), the quadratic equation cannot intersect the \(x\)-axis in the real number plane, meaning it doesn't have real solutions.Here's how you can handle complex roots:

• Recognize that the solutions will involve the imaginary unit \(i\), where \(i^2 = -1 \).
• The roots are expressed in the form \(x = \frac{-b \pm i\sqrt{|b^2 - 4ac|}}{2a} \), which simplifies to \(x = \frac{-b}{2a} \pm i \frac{\sqrt{|b^2 - 4ac|}}{2a} \).

This form clearly shows the real part \(\frac{-b}{2a} \) and the imaginary part \( \pm i \frac{\sqrt{|b^2 - 4ac|}}{2a} \). Complex conjugate roots occur in pairs and are vital in many fields, especially in signal processing and advanced geometry. Understanding complex roots enriches your overall grasp of mathematics by extending real number systems to complex planes.