Problem 59 - Algebra for College Students

Question

For each of the following quadratic functions, use the discriminant to determine the number of real-number zeros, and then graph the function with a graphing calculator to check your answer. (a) \(f(x)=3 x^{2}-15 x-42\) (b) \(f(x)=2 x^{2}-36 x+162\) (c) \(f(x)=-4 x^{2}-48 x-144\) (d) \(f(x)=2 x^{2}+2 x+5\) (e) \(f(x)=4 x^{2}-4 x-120\) (f) \(f(x)=5 x^{2}-x+4\)

Step-by-Step Solution

Answer

Functions (a), (b), and (e) have 2 real zeros; (c) and (f) have no real zeros; (d) has 2 complex roots.

Step 1: Identify the Quadratic Equation Coefficients

For each quadratic function, identify the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ax^2 + bx + c\).

Step 2: Calculate the Discriminant

For each function, calculate the discriminant using the formula \(\Delta = b^2 - 4ac\). This value will help determine the number of real-number zeros.

Step 3: Determine the Number of Real-Number Zeros

Using the discriminant value calculated earlier, determine the number of real-number zeros:- If \(\Delta > 0\), there are 2 distinct real zeros.- If \(\Delta = 0\), there is 1 real zero (a repeated root).- If \(\Delta < 0\), there are no real zeros (2 complex roots).

Step 4: Verify Using a Graphing Calculator

Use a graphing calculator to graph each quadratic function. Check the graph for x-intercepts, which represent the real-number zeros found using the discriminant.

Key Concepts

Discriminant

When tackling quadratic equations, the discriminant is a key tool for understanding the nature of the equation's solutions. To find the discriminant, you use the formula \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the equation \( ax^2 + bx + c \). The discriminant helps reveal important information about the roots of the quadratic equation.

• If \( \Delta > 0 \), the equation has two distinct real-number zeros. This means our quadratic equation will cross the x-axis at two points.
• If \( \Delta = 0 \), the equation has exactly one real-number zero, indicating that the parabola touches the x-axis at a single point (also known as a repeated root).
• If \( \Delta < 0 \), the real-number zeros do not exist, and instead, the equation has two complex roots.

By calculating the discriminant, you can quickly assess the type of solutions you will be working with, which sets the stage for deeper analysis.

Real-Number Zeros

Real-number zeros are the points where a quadratic equation intersects the x-axis on a graph. These zeros can be found by solving the quadratic equation or by examining the discriminant.
- When \( \Delta > 0 \), the equation has two real-number zeros, meaning the parabola crosses the x-axis at two distinct points.- If \( \Delta = 0 \), we have one real number zero — also known as a double root or repeated root — where the parabola just touches the axis.
Finding these zeros is useful in understanding the behavior of the curve and provides valuable insights into the solutions of the equation.

Graphing Calculator

A graphing calculator is a powerful tool when exploring quadratic equations because it visually displays their behavior.
To check the real-number zeros found using the discriminant, plot the equation on a graphing calculator. You'll look for x-intercepts as these represent the real-number zeros. Here's how to use this tool:

• Enter the quadratic equation into the graphing function of your calculator.
• Observe where the graph intersects the x-axis. These intersections are the zeros or roots of the equation.
• Use the x-intercepts found visually on the graph to double-check your discriminant calculations.

This process not only confirms theoretical calculations but also enhances your understanding by providing a visual representation of the equation.

Complex Roots

When a quadratic equation has a negative discriminant (\( \Delta < 0 \)), it means the equation has no real-number zeros and instead has complex roots. Unlike real-number zeros, these roots do not intersect the x-axis on the graph.
Complex roots occur in conjugate pairs and are typically expressed in the form \( a \pm bi \), where \( i \) represents the square root of -1. This occurs because the solutions involve the square root of a negative number, introducing the imaginary unit.

• Complex roots are shown as points not on the real number line.
• These roots are essential for certain advanced topics and contexts within mathematics, signifying solutions that extend beyond the real plane.

Understanding complex roots is crucial for a comprehensive grasp of quadratic functions and their broader implications in mathematical equations.