Problem 48 - Calculus with Concepts in Calculus

Question

Approximate all zeros of the function to the nearest hundredth. Let \(f(x)=a x^{2}+b x+c\) with \(a \neq 0\). Suppose \(f\) has two zeros, \(z_{1}\) and \(z_{2}\). Express \(z_{1}+z_{2}\) in terms of \(a, b\), and \(c\).

Step-by-Step Solution

Answer

The sum of the zeros, \(z_1 + z_2\), is \(-\frac{b}{a}\).

Step 1: Understand the Quadratic Formula

To find the zeros of the quadratic function \(f(x) = ax^2 + bx + c\), we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula finds the values of \(x\) where \(f(x)\) is zero.

Step 2: Identify the Zeros

In the quadratic formula, the zeros are represented by \(z_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\) and \(z_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\). These are the values of \(x\) for which \(f(x) = 0\).

Step 3: Sum of the Zeros

To find the sum of the zeros \(z_1 + z_2\), add the expressions for \(z_1\) and \(z_2\): \[ z_1 + z_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a} \].

Step 4: Simplify the Expression

Combine the fractions: \[ z_1 + z_2 = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a} \]. The terms involving the square roots cancel out, leaving \[ z_1 + z_2 = \frac{-2b}{2a} \].

Step 5: Final Simplification

Simplify the expression \(\frac{-2b}{2a}\) to \(-\frac{b}{a}\). This is the sum of the zeros expressed in terms of \(a\) and \(b\).

Key Concepts

The Quadratic Formula

The quadratic formula is a powerful tool in mathematics for finding the zeros of a quadratic function, which is a function defined by the expression \(f(x) = ax^2 + bx + c\), where \(a eq 0\). A quadratic function forms a parabola on a graph and can have up to two real solutions, known as zeros. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]It consists of three main parts:

• \(-b\): This flips the sign of \(b\).
• \(\pm\): This indicates that there are typically two solutions; one for addition and one for subtraction.
• \(\sqrt{b^2 - 4ac}\): Known as the discriminant, it determines the nature of the solutions. If it's positive, there are two distinct real solutions. If zero, one real solution exists, and if negative, no real solutions.

The quadratic formula provides a complete solution to any quadratic equation, allowing us to find the points where the function intersects the x-axis, also known as the zeros.

Zeros of a Function

Zeros of a function are the values of \(x\) that make the function equal to zero. In other words, they are the points where the graph of the function crosses or touches the x-axis. For a quadratic function \(f(x) = ax^2 + bx + c\), the zeros can be found using the quadratic formula:

• \(z_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\)
• \(z_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\)

These solutions are derived by setting \(f(x) = 0\) and solving for \(x\). The zeros represent the solutions of the quadratic equation, indicating where the function has no value (zero). Depending on the discriminant \(b^2 - 4ac\), these zeros can be real or complex numbers, but in cases where they are real, they can be plotted directly as the x-intercepts of the parabola on a Cartesian plane.

Sum of Zeros

The sum of the zeros of a quadratic function \(f(x) = ax^2 + bx + c\) provides valuable information about the equation. This concept stems from the properties of the quadratic formula. If we have zeros \(z_1\) and \(z_2\), their sum can be expressed without solving the quadratic formula straightforwardly.Starting with the quadratic formula solutions:

• \(z_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\)
• \(z_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\)

Adding these gives:\[ z_1 + z_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]Upon simplification, the term \(\sqrt{b^2 - 4ac}\) cancels out, resulting in:\[z_1 + z_2 = \frac{-2b}{2a} = -\frac{b}{a}\]This relationship demonstrates that the sum of the zeros does not depend on the discriminant but only on the coefficients \(a\) and \(b\). This property is based on Viète's formulas, which link the coefficients of polynomials to sums and products of their roots, making it a fundamental aspect of algebra and quadratic identities.