Problem 6 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(5 x^{2}+2 x-10=0\)

Step-by-Step Solution

Answer

Sum: \(-\frac{2}{5}\), Product: \(-2\).

Step 1: Understand Vieta's Formulas

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

Step 2: Identify Coefficients

In the given equation \(5x^2 + 2x - 10 = 0\), identify the coefficients: \(a = 5\), \(b = 2\), and \(c = -10\).

Step 3: Calculate the Sum of the Roots

Using Vieta's formula for the sum of the roots, \(r_1 + r_2 = -\frac{b}{a}\). Substitute the coefficients into the formula: \(r_1 + r_2 = -\frac{2}{5}\). This means the sum of the roots is \(-\frac{2}{5}\).

Step 4: Calculate the Product of the Roots

Using Vieta's formula for the product of the roots, \(r_1 \cdot r_2 = \frac{c}{a}\). Substitute the coefficients into the formula: \(r_1 \cdot r_2 = \frac{-10}{5} = -2\). This means the product of the roots is \(-2\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the standard form: \( ax^2 + bx + c = 0 \). The key components of this equation are the coefficients \(a\), \(b\), and \(c\), which are constants. The variable \(x\) is the unknown we aim to solve for. Each part of the quadratic equation influences its shape when plotted on a graph, forming a parabola.

• The coefficient \(a\) determines the parabola's direction. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• The coefficient \(b\) affects the parabola's horizontal placement.
• The constant \(c\) determines where the parabola crosses the y-axis.

Quadratic equations can be solved using various methods like factoring, completing the square, or using the quadratic formula. However, understanding the relationships of coefficients through Vieta’s formulas can offer quick insights into the roots of the equation without directly solving it.

Exploring Roots of a Polynomial

Roots of a polynomial are the solutions to the equation \( ax^2 + bx + c = 0 \), where for quadratic equations, they are often termed as the 'solutions' or 'zeros.'

• These roots represent the x-values where the graph of the polynomial intersects the x-axis.
• In a quadratic equation, there might be two real roots, one real root, or no real roots, depending on the discriminant \(b^2 - 4ac\).

For example, in the equation \(5x^2 + 2x - 10 = 0\), the roots are the \(x\) values that make the equation true when substituted back into the expression. However, using Vieta’s formulas can help us understand the nature of these roots without directly finding their values.

Sum and Product of Roots with Vieta's Formulas

Vieta's formulas provide a direct link between a quadratic equation's coefficients and its roots, enabling predictions about the sum and product of these roots without solving the entire equation.

• The sum of the roots, \(r_1 + r_2\), of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \[-\frac{b}{a}\]. This indicates the sum is dependent on the coefficients \(b\) and \(a\).
• The product of the roots, \(r_1 \cdot r_2\), is found using: \[\frac{c}{a}\]. Here, both coefficients \(c\) and \(a\) determine the product.

For instance, in the equation \(5x^2 + 2x - 10 = 0\):- The sum of the roots is \[-\frac{2}{5}\], meaning that when you add the roots together, you get this result.- The product of the roots is \[-2\]. Multiplying the roots produces this value.Understanding and applying Vieta’s formulas offer a quick way to examine essential properties of a quadratic equation’s roots without detailed computations.