Problem 8 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(-x^{2}+3 x+1=0\)

Step-by-Step Solution

Answer

Sum of the roots is 3; product of the roots is -1.

Step 1: Identifying the Coefficients

For the quadratic equation \[-x^2 + 3x + 1 = 0\] we identify the coefficients as follows:- \(a = -1\)- \(b = 3\)- \(c = 1\)

Step 2: Using Vieta’s Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \(ax^2 + bx + c = 0\):- The sum of the roots \((r_1 + r_2)\) is given by \(-\frac{b}{a}\).- The product of the roots \((r_1 \cdot r_2)\) is given by \(\frac{c}{a}\).

Step 3: Calculating the Sum of the Roots

Using Vieta’s formula for the sum of the roots:\[r_1 + r_2 = -\frac{b}{a} = -\frac{3}{-1} = 3\]Thus, the sum of the roots is 3.

Step 4: Calculating the Product of the Roots

Using Vieta’s formula for the product of the roots:\[r_1 \cdot r_2 = \frac{c}{a} = \frac{1}{-1} = -1\]Thus, the product of the roots is -1.

Key Concepts

Sum of the Roots

When exploring quadratic equations, one of the most interesting aspects is how the sum of the roots is connected to the equation's coefficients. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas provide a way to find the sum of its roots without solving it. According to Vieta's formulas, the sum of the roots \(r_1 + r_2\) can be determined by the formula \(-\frac{b}{a}\), where \(b\) is the coefficient of \(x\) and \(a\) is the coefficient of \(x^2\). This means you simply divide the negative of the \(b\) coefficient by the \(a\) coefficient to find the sum.

For example, in the quadratic equation \(-x^2 + 3x + 1 = 0\), \(a = -1\) and \(b = 3\). Applying Vieta's formula, we get:

• Sum of the roots: \( -\frac{3}{-1} = 3 \)

This indicates the sum of the roots is 3, illustrating how Vieta's formulas provide a straightforward approach to determining the sum without any solving process.

Product of the Roots

Just like the sum, the product of the roots of a quadratic equation is directly linked to the coefficients of the equation through Vieta's formulas. Specifically, for the equation \( ax^2 + bx + c = 0 \), the formula to find the product of its roots \(r_1 \cdot r_2\) is \(\frac{c}{a}\), where \(c\) is the constant term and \(a\) is the coefficient of \(x^2\).

This relationship highlights a fascinating symmetry in quadratic equations, where their entire essence (in terms of the roots' sum and product) can be captured in terms of the equation's coefficients.

Consider again our example equation \(-x^2 + 3x + 1 = 0\), where \(a = -1\) and \(c = 1\). Using Vieta's formula for the product gives:

• Product of the roots: \( \frac{1}{-1} = -1 \)

Thus, the product of the roots is -1. This offers a neat shortcut for deducing the product without completing the actual solving of the equation.

Quadratic Equation

Quadratic equations are a cornerstone of algebra and appear in various real-world applications. A quadratic equation is generally presented in the standard form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). It represents a second-degree polynomial, indicating it will have two solutions, or roots. These roots can be real or complex numbers, depending on the discriminant value \(b^2 - 4ac\).

Quadratic equations are primarily solved using methods such as factoring, completing the square, or the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). However, Vieta’s formulas offer elegant shortcuts for discovering certain relationships about the roots without solving the entire equation.

These formulas reveal much about quadratic equations’ structure, showing how the sum and product of its roots are surprisingly interlinked with the coefficients themselves. Thanks to their simplicity and power, quadratic equations and their properties, such as through Vieta’s insight, are vital topics in mathematics and educational curricula.