Problem 5 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

Sum of roots: \(\frac{3}{2}\), Product of roots: \(-1\).

Step 1: Identify the coefficients

For the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 2\), \(b = -3\), and \(c = -2\).

Step 2: Apply the sum of roots formula

Use the formula for the sum of the roots of a quadratic equation: \( \text{Sum of roots} = -\frac{b}{a} \). Substitute in the values, \(-\frac{-3}{2}\).

Step 3: Calculate the sum of the roots

Calculate \(-\frac{-3}{2}\), which simplifies to \(\frac{3}{2}\). Therefore, the sum of the roots is \(\frac{3}{2}\).

Step 4: Apply the product of roots formula

Use the formula for the product of the roots of a quadratic equation: \( \text{Product of roots} = \frac{c}{a} \). Substitute in the values, \(\frac{-2}{2}\).

Step 5: Calculate the product of the roots

Calculate \(\frac{-2}{2}\), which simplifies to \(-1\). Therefore, the product of the roots is \(-1\).

Key Concepts

Understanding the Sum of Roots in Quadratic Equations

Quadratic equations are often expressed in the form, \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. One fundamental concept to grasp when dealing with these equations is the sum of their roots. The roots of a quadratic equation are the values of \(x\) that satisfy the equation. To find the sum of these roots, we use a formula derived from the standard form of the quadratic. It is given by:

• \(\text{Sum of roots} = -\frac{b}{a}\)

This formula offers a straightforward way to determine the sum of the roots without solving the entire equation. In the equation \(2x^2 - 3x - 2 = 0\), for instance, we identify \(a = 2\) and \(b = -3\). Plugging into the formula, we calculate \(-\frac{-3}{2}\), simplifying to \(\frac{3}{2}\). This value represents the sum of the roots. Knowing this formula allows you to quickly find the sum whenever you know the coefficients \(a\) and \(b\).
Understanding this concept can save time and effort in solving quadratic equations.

How to Determine the Product of Roots

The product of roots is another key aspect when analyzing quadratic equations. Remembering the roots are solutions to \(ax^2 + bx + c = 0\), the product lets us know if multiplying them gives us a meaningful number. The formula to determine this product is as follows:

• \(\text{Product of roots} = \frac{c}{a}\)

Identifying the coefficients in our equation \(2x^2 - 3x - 2 = 0\), we have \(a = 2\) and \(c = -2\). Plugging these into the formula gives \(\frac{-2}{2}\), which simplifies to \(-1\). This value indicates whether the roots multiply to a positive or negative result. Hence, without solving for the individual roots, we determine that the product is \(-1\).
Recognizing this pattern greatly aids in understanding the nature of the roots in relation to the coefficients.

Vieta's Formulas for Quadratic Equations

Named after the French mathematician François Viète, Vieta's formulas provide a beautiful relationship between the coefficients of a polynomial and its roots. For quadratic equations, these formulas give us a concise way to find the sum and product of the roots without delving into messy calculations or extra steps. They state:

• The sum of the roots \( = -\frac{b}{a}\)
• The product of the roots \( = \frac{c}{a}\)

These formulas summarize important properties of the roots in relation to their coefficients. They are especially helpful in quickly making sense of the solutions to quadratic equations, as evidenced by our work with \(2x^2 - 3x - 2 = 0\). Here, using Vieta's, we determine the sum to be \(\frac{3}{2}\) and the product to be \(-1\).
Vieta's formulas are a valuable tool to have, making it easier and quicker to unravel key characteristics of quadratic roots.