Problem 66 - Algebra 2

Question

The two zeros of a quadratic function are labeled \(x_{1}\) and \(x_{2}\) on the graph. Which expression has the greatest value? A. 2\(x_{1}\) B. \(x_{2}\) C. \(x_{2}-x_{1}\) D. \(x_{2}+x_{1}\)

Step-by-Step Solution

Answer

D: \(x_{2} + x_{1}\) typically has the greatest value, especially if both zeros are positive.

Step 1: Understand the Problem

We need to determine which of the given expressions involving the zeros of a quadratic function yields the greatest value. The zeros are labeled as \(x_{1}\) and \(x_{2}\), and the expressions are 2\(x_{1}\), \(x_{2}\), \(x_{2}-x_{1}\), and \(x_{2}+x_{1}\).

Step 2: Review Properties of Zeros

Recall that the zeros \(x_{1}\) and \(x_{2}\) of a quadratic function may differ, depending on the orientation of the parabola and given the symmetry, one value could be greater or less than the other.

Step 3: Evaluate Expression A: 2\(x_{1}\)

Expression 2\(x_{1}\) represents the value of zero \(x_{1}\) doubled. If \(x_{1}\) is negative or small, this value might not be large. Compare \(x_{1}\) to other expressions.

Step 4: Evaluate Expression B: \(x_{2}\)

Expression \(x_{2}\) reflects the value of the second zero. It holds its own natural value and can either support a larger, smaller, or equal result when compared.

Step 5: Evaluate Expression C: \(x_{2} - x_{1}\)

Expression \(x_{2} - x_{1}\) represents the difference between the two zeros. If \(x_{2}\) is greater than \(x_{1}\), this value will be positive.

Step 6: Evaluate Expression D: \(x_{2} + x_{1}\)

Expression \(x_{2} + x_{1}\) is simply the sum of the two zeros. In cases where one zero is negative, the sum will be reduced.

Step 7: Consider Possible Scenarios or Value Relations

Consider scenarios: If both \(x_{1}\) and \(x_{2}\) are positive and \(x_{2} > x_{1}\), \(x_{2} + x_{1}\) will exceed any doubled zero value. If \(x_{1}\) is negative, \(x_{2} - x_{1}\) could be large. Evaluate each based on probable limits and real cases.

Step 8: Conclusion Based on Scenario Analysis

Based on step-by-step assessment, the sum \(x_{2} + x_{1}\) typically results in a greater combined value under normal root conditions (i.e. positive or one positive, other negative). While specific cases may differ, generally across symmetric, typical quadratic functions, this sum should be greatest.

Key Concepts

Zeros of a Quadratic

The zeros of a quadratic function are the values of \(x\) where the quadratic equation equals zero. These are also the points where the graph of the quadratic function, a parabola, intersects the x-axis. Identifying the zeros helps understand the characteristics of the quadratic equation. The standard form of a quadratic function is \(ax^2 + bx + c = 0\). By using the quadratic formula:

• \(x_{1}, x_{2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

we can find the zeros \(x_{1}\) and \(x_{2}\).

Once we find these zeros, it is important to analyze their significance. For example, they tell us about the root characteristics, which are essential in many mathematical problems and graphs. Keep in mind, the zeros not only represent intersections with the x-axis but also influence the vertex and direction of the parabola.

Sum of Roots

Understanding the sum of roots is critical in dealing with quadratic equations. The sum of the roots is commonly expressed through Viète's formulas. For the quadratic function \(ax^2 + bx + c = 0\), the relation between the coefficients and the sum of roots is:

• \(x_{1} + x_{2} = -\frac{b}{a}\)

This expression reveals that regardless of how the roots themselves appear, their combined sum is influenced by the coefficients of the equation. This is particularly useful in forms beyond numerical solutions as it reveals direct relationships between different parts of the quadratic equation.

In the context of identifying which expression from the exercise holds the greatest value, understanding this sum helps unravel the comparative analysis, often indicating how the zeros are working together rather than individually.

Properties of Roots

When it comes to quadratic equations, the properties of roots shed light on how they behave and interact, both mathematically and graphically. The roots are interconnected by several mathematical properties, such as their product and sum, which help in solving quadratics without finding exact values.

### Deep Dive into Properties

• **Product of Roots:** The product \(x_{1} \cdot x_{2} = \frac{c}{a}\) of the quadratic \(ax^2 + bx + c = 0\) gives insight into their multiplicative nature.
• **Symmetry:** Quadratic functions exhibit symmetry around the vertex, making one root a direct reflection across this axis depending on whether the parabola opens upward or downward.
• **Sign Characteristics:** The nature of the discriminant \(b^2 - 4ac\) determines whether roots are real or complex, and whether they are distinct or equal, further influencing the characteristics of the sum and product.

In practical terms, knowing these relationships reduces solving complex equations into manageable algebraic manipulations, providing clarity in situations involving various expressions of roots, such as in the original exercise.