Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-14\) : a. For each given value of the discriminant of a quadratic equation with rational coefficients, determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use your answer to part a to determine the number of \(x\) -intercepts of the graph of the corresponding quadratic function. -4

Step-by-Step Solution

Answer

The roots are not real; 0 x-intercepts.

Step 1: Understanding the Discriminant

The discriminant of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). It determines the nature of the roots of the quadratic equation.

Step 2: Analyzing the Given Discriminant

We are given the discriminant \( \Delta = -4 \). A negative discriminant (\( \Delta < 0 \)) means the roots of the quadratic equation are not real numbers; they are complex or imaginary. Therefore, for \( \Delta = -4 \), the roots are complex and not real.

Step 3: Relating to Graph Intercepts

A quadratic equation produces a parabola when graphed. If the roots are not real numbers, the parabola does not cross the x-axis. Therefore, the number of x-intercepts is 0.

Key Concepts

Nature of Roots

In a quadratic equation, the discriminant plays a crucial role in determining the nature of its roots. The discriminant is a part of the quadratic formula, symbolized as \( \Delta = b^2 - 4ac \). Below are the implications of various values of the discriminant on the roots of a quadratic equation:

• If \( \Delta > 0 \), the roots are real and distinct. In this case:
  • If \( \Delta \) is a perfect square, the roots are rational and unequal.
  • If \( \Delta \) is not a perfect square, the roots are irrational and unequal.
• If \( \Delta = 0 \), both roots are real and equal, which means they are rational and match each other.
• If \( \Delta < 0 \), the roots are not real. Instead, they are complex or imaginary, occurring as complex conjugates.

The nature of the roots is extremely helpful when examining a quadratic’s characteristics, including its graph and symmetry.

Quadratic Function Graph

A quadratic equation, when plotted on a graph, produces a U-shaped curve known as a parabola. The x-intercepts or roots are the points where the parabola crosses the x-axis. Understanding the discriminant helps in predicting these intercepts:

• If \( \Delta > 0 \), the parabola crosses the x-axis at two distinct points, indicating two real and unequal roots.
• If \( \Delta = 0 \), the parabola touches the x-axis at exactly one point, implying a repeated real root (also known as a double root).
• If \( \Delta < 0 \), the parabola does not intersect the x-axis at all, indicating the absence of real roots as the solutions are complex.

The shape and direction of the parabola are dictated by the leading coefficient \( a \). If \( a > 0 \), the parabola opens upwards. Conversely, if \( a < 0 \), it opens downwards. Recognizing these x-intercepts can give insights into the real-world applicability and graph-related questions.

Complex Roots

Complex roots in the context of quadratics arise when the discriminant \( \Delta \) is negative. These roots are not on the real number line and involve imaginary numbers, expressed as \( \pm i \sqrt{-\Delta} \) where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

• Complex roots always occur in conjugate pairs, meaning if one root is \( a + bi \), the other is \( a - bi \).
• The presence of complex roots implies that the quadratic graph (the parabola) lies entirely above or below the x-axis, depending on the direction in which it opens.
• Though they don't represent real intersections on the Cartesian plane, complex roots are essential in various fields such as engineering and physics, where they help to describe oscillations and wave behaviors.

Understanding when roots are complex helps in scenarios where real-life modeling and signal analysis are necessary. It emphasizes the diversity and applicability of quadratics beyond mere real-number solutions.