Problem 7 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(3 x^{2}-6 x+4=0\)

Step-by-Step Solution

Answer

Sum of roots: 2, Product of roots: \(\frac{4}{3}\).

Step 1: Recognize the Polynomial Form

The given equation \(3x^2 - 6x + 4 = 0\) is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = -6\), and \(c = 4\).

Step 2: Apply Vieta's Formulas

Vieta's formulas tell us that for the quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(r_1 + r_2 = -\frac{b}{a}\) and the product of the roots \(r_1 \cdot r_2 = \frac{c}{a}\).

Step 3: Calculate the Sum of the Roots

Using the formula \(r_1 + r_2 = -\frac{b}{a}\), substitute \(b = -6\) and \(a = 3\): \[-(-6) / 3 = \frac{6}{3} = 2\] Thus, the sum of the roots is 2.

Step 4: Calculate the Product of the Roots

Using the formula \(r_1 \cdot r_2 = \frac{c}{a}\), substitute \(c = 4\) and \(a = 3\): \[\frac{4}{3}\] So, the product of the roots is \(\frac{4}{3}\).

Key Concepts

Understanding Quadratic Equations

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are called 'quadratic' because they are associated with "quad", a prefix denoting "square" in mathematics. Quadratic equations represent parabolas when graphed on a coordinate plane.
For example, in the equation \(3x^2 - 6x + 4 = 0\), the coefficients are \(a = 3\), \(b = -6\), and \(c = 4\). The solutions to these equations, also known as the 'roots', are the values of \(x\) that make the equation true. These can be found via factoring, completing the square, or using the quadratic formula, though Vieta’s formulas offer a quick insight into the sum and product of roots without solving it fully.

Exploring the Sum of the Roots

The sum of the roots of a quadratic equation is a concept derived from Vieta’s formulas. For any quadratic equation in the standard form \(ax^2 + bx + c = 0\), the sum of its roots \((r_1 + r_2)\) can be calculated directly using the formula:

• \(r_1 + r_2 = -\frac{b}{a}\)

This formula emerges because the quadratic equation can be expressed as \(a(x - r_1)(x - r_2)\). Here, the term \(-b/a\) represents the sum of the roots due to the symmetry in the factors. It essentially comes from considering the expansion of \((x - r_1)(x - r_2)\), which gives \(x^2 - (r_1 + r_2)x + r_1r_2\).

For our given quadratic equation \(3x^2 - 6x + 4 = 0\), we substitute \(b = -6\) and \(a = 3\) into the formula to find the sum of the roots:
\(r_1 + r_2 = -\frac{-6}{3} = \frac{6}{3} = 2\). Hence, the sum of the roots is 2. This provides a neat and efficient method to understand the relationship between the roots without necessarily finding them individually.

Understanding the Product of the Roots

The product of the roots offers insight into another aspect of quadratic equations. According to Vieta’s formulas, for a quadratic equation \(ax^2 + bx + c = 0\), the product of its roots \((r_1 \cdot r_2)\) is given by:

• \(r_1 \cdot r_2 = \frac{c}{a}\)

This formula reflects the constant term \(c\) in relation to the leading coefficient \(a\), from the expanded form \( (x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1r_2 \).
In the example equation \(3x^2 - 6x + 4 = 0\), substituting \(c = 4\) and \(a = 3\) gives us:
\(r_1 \cdot r_2 = \frac{4}{3}\).
The product of the roots thus turns out to be \(\frac{4}{3}\). This product of the roots is an important characteristic and helps us understand the nature of the roots even if they are complex numbers or irrational in nature.