Problem 44 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. Without drawing the graph or completely solving the equation, explain how to find the number of \(x\) -intercepts of a quadratic function.

Step-by-Step Solution

Answer

Calculate the discriminant: if it's positive, there are two x-intercepts; if zero, one x-intercept; if negative, no x-intercepts.

Step 1: Review the Quadratic Function

The standard form of a quadratic function is given by \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The \( x \)-intercepts of the function are the values of \( x \) where \( f(x) = 0 \).

Step 2: Understand the Role of the Discriminant

The discriminant, \( \Delta \), of the quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). The value of the discriminant helps determine the nature and number of \( x \)-intercepts (real roots) of the quadratic function.

Step 3: Determine the Number of x-Intercepts Using the Discriminant

Assess the value of the discriminant \( \Delta \):- If \( \Delta > 0 \), the quadratic function has two distinct real \( x \)-intercepts.- If \( \Delta = 0 \), the quadratic function has exactly one real \( x \)-intercept (a repeated root).- If \( \Delta < 0 \), the quadratic function has no real \( x \)-intercepts, as the roots are complex.

Key Concepts

X-intercepts

In mathematics, the term x-intercepts refers to the points where a graph crosses the x-axis. For quadratic functions, these points are particularly important because they represent the solutions or roots of the equation when the function equals zero. Finding the x-intercepts of a quadratic function gives us insight into where the function reaches zero on the x-axis.

To identify the x-intercepts, you set the quadratic equation in standard form, which is given by:

• \( ax^2 + bx + c = 0 \)

This equation is known as the standard equation of a quadratic function.

• The solutions to this equation, if any, are the x-intercepts.

These are the real values of \(x\) that satisfy the equation when \(f(x) = 0\). The calculation of these solutions involves understanding various components of the quadratic equation, including the discriminant.

Discriminant

The discriminant is a key part of the quadratic formula, and it is crucial for understanding how many x-intercepts or real roots a quadratic function might have. The discriminant is represented by the symbol \(\Delta\) and is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

This value is derived from the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). The discriminant provides essential insights about the nature of the roots of the quadratic equation. Depending on the value of the discriminant, we can determine the number of x-intercepts of the function:

• If \(\Delta > 0\), the function has two distinct real x-intercepts.
• If \(\Delta = 0\), there is exactly one real x-intercept, implying a repeated root.
• If \(\Delta < 0\), the function has no real x-intercepts, which means the roots are complex numbers.

Evaluating the discriminant is a vital step in determining the behavior and solutions of quadratic functions.

Real Roots

Real roots are the solutions of the quadratic equation that are real numbers. These solutions are essential as they tell us precisely where a quadratic function touches or crosses the x-axis. Real roots and x-intercepts are fundamentally connected because real roots indicate the presence and location of x-intercepts.

When we say a quadratic equation has real roots, it means that upon solving the equation \(ax^2 + bx + c = 0\), we obtain real number solutions that satisfy this equation.

• The nature of these roots—whether they are distinct or repeated—is determined by the discriminant.

The discriminant communicates how many x-intercepts we have, leading us to the depth of the function's interaction with the x-axis. Thus, understanding real roots is crucial for solving quadratic functions and knowing precisely the number and nature of x-intercepts.

Standard Form of Quadratic Function

The standard form of a quadratic function is essential because it provides a straightforward way to express and analyze quadratics. The standard form is:

• \( f(x) = ax^2 + bx + c \)

Here, \(a\), \(b\), and \(c\) are constant coefficients, and \(a eq 0\). Why must \(a\) not be zero? Because if it were, the equation would become linear instead of quadratic.

• This form allows us to easily identify the axis of symmetry, vertex, and x-intercepts of the quadratic graph.
• It provides the base structure from which the discriminant and other important calculations are derived.

Understanding this form helps us systematically address the properties of the quadratic function, including finding its x-intercepts and analyzing its graph.