Problem 16 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

Sum of roots is -2, and product of roots is -3.

Step 1: Identifying the Quadratic Formula Parameters

A quadratic equation is generally formatted as \( ax^2 + bx + c = 0 \). In the given equation \( x^2 + 2x - 3 = 0 \), we can identify \( a = 1 \), \( b = 2 \), and \( c = -3 \). This identification will help us apply the properties of quadratic equations to find the sum and product of the roots.

Step 2: Applying Vieta's Formulas for the Sum of the Roots

According to Vieta's formulas, for a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) is given by \(-\frac{b}{a}\). Substituting \( a = 1 \) and \( b = 2 \), we get the sum as \(-\frac{2}{1} = -2\).

Step 3: Applying Vieta's Formulas for the Product of the Roots

Again, using Vieta's formulas, the product of the roots \( \alpha \times \beta \) for the quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \frac{c}{a} \). Substituting \( a = 1 \) and \( c = -3 \), we find the product as \( \frac{-3}{1} = -3 \).

Key Concepts

Vieta's Formulas

Vieta's Formulas are a set of relationships between the coefficients of a polynomial and sums and products of its roots. They were named after the French mathematician François Viète. In the context of a quadratic equation formatted as \( ax^2 + bx + c = 0 \), Vieta's Formulas provide a quick way to find both the sum and the product of the roots without actually solving the equation.

For a quadratic equation:

• The sum of the roots is given by \( -\frac{b}{a} \).
• The product of the roots is given by \( \frac{c}{a} \).

These formulas are invaluable when dealing with quadratic problems as they simplify computations significantly. For example, in the equation \( x^2 + 2x - 3 = 0 \), we can quickly determine that the sum of the roots is \( -2 \) and the product is \( -3 \). This is achieved by plugging in the coefficients \( a = 1 \), \( b = 2 \), and \( c = -3 \) into the formulas.

Sum of the Roots

The sum of the roots of a quadratic equation is a handy piece of information that can be derived directly using Vieta's Formulas. The formula \( -\frac{b}{a} \) indicates the sum of the roots \( \alpha + \beta \) for any quadratic equation \( ax^2 + bx + c = 0 \).

Here’s how it works:
By simply substituting the values from the quadratic equation, we can find the sum without needing to solve it. For the equation \( x^2 + 2x - 3 = 0 \), we find:
\( a = 1 \), \( b = 2 \), resulting in:

• Sum of roots: \( -\frac{2}{1} = -2 \).

This intuitively implies that if you were to add together the two solutions to the equation, you would arrive at \( -2 \). This method saves time and simplifies calculations, especially useful in exams or timed settings.

Product of the Roots

Finding the product of the roots using Vieta’s Formulas is as straightforward as finding their sum. For the quadratic equation \( ax^2 + bx + c = 0 \), the formula for the product of the roots \( \alpha \times \beta \) is \( \frac{c}{a} \).

Let's consider the quadratic equation \( x^2 + 2x - 3 = 0 \) again:
Identifying the parameters \( a = 1 \) and \( c = -3 \):

• Product of roots: \( \frac{-3}{1} = -3 \).

This means if you multiply the two roots together, you get \( -3 \). This powerful approach allows us to grasp the nature of the roots without actually finding their exact values, making it highly efficient in handling polynomial equations.

Quadratic Formula Parameters

Understanding the parameters in the quadratic formula is crucial for utilizing Vieta's Formulas effectively. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In this equation, \( a \), \( b \), and \( c \) are coefficients where \( a eq 0 \).

The parameters have specific roles:

• \( a \): the coefficient of \( x^2 \), determines the opening direction and width of the parabola represented by the quadratic function.
• \( b \): the coefficient of \( x \), affects the position of the axis of symmetry of the graph.
• \( c \): the constant term, gives the \( y \)-intercept of the graph.

Understanding these coefficients allows us to apply Vieta's Formulas effectively. For instance, in the equation \( x^2 + 2x - 3 = 0 \), \( a = 1 \), \( b = 2 \), and \( c = -3 \). These parameters are key to solving and understanding the equation's properties without solving the equation fully.