Problem 4 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(x^{2}+4 x+5=0\)

Step-by-Step Solution

Answer

The sum of the roots is -4, and the product is 5.

Step 1: Identifying Coefficients

For the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are \( a = 1 \), \( b = 4 \), and \( c = 5 \).

Step 2: Formula for Sum of the Roots

The formula to find the sum of the roots \( \alpha + \beta \) of a quadratic equation is given by \( -\frac{b}{a} \). Using the coefficients from the equation, we have \( -\frac{4}{1} = -4 \).

Step 3: Formula for Product of the Roots

The formula to find the product of the roots \( \alpha \cdot \beta \) is given by \( \frac{c}{a} \). Using the coefficients from the equation, we have \( \frac{5}{1} = 5 \).

Key Concepts

Identifying Coefficients in Quadratic Equations

In the world of quadratic equations, the coefficient identification is your starting point. Consider the general form of a quadratic equation: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are numbers called coefficients:

• \( a \) is the coefficient of \( x^2 \), known as the quadratic coefficient. It's also viewed as the leading coefficient since it dictates the "width" of the parabola.
• \( b \) is the linear coefficient, attached to \( x \). It affects the placement and direction the parabola opens.
• \( c \) is the constant term, which gives the equation its vertical shift on a graph.

Recognizing these coefficients accurately is key. For example, in the equation \( x^2 + 4x + 5 = 0 \), \( a = 1 \), \( b = 4 \), and \( c = 5 \). These numbers will be essential in further calculations, such as finding roots.

Sum of Roots in Quadratic Equations

The sum of the roots of a quadratic equation is a concept that stems from Vieta's formulas. It allows us to find the sum without actually calculating the roots themselves. For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of its roots, denoted by \( \alpha + \beta \), follows a simple formula:

\[\alpha + \beta = -\frac{b}{a}\]
Given our equation \( x^2 + 4x + 5 = 0 \), plug the coefficients into the formula: \(-\frac{4}{1} = -4\).

This means that the sum of the roots is \(-4\). This allows you to understand important characteristics of the equation just by examining its coefficients. You don't need to actually find \( \alpha \) and \( \beta \) to know their sum.

Product of Roots in Quadratic Equations

Continuing with Vieta's formulas, the product of the roots of a quadratic equation is equally accessible. For \( ax^2 + bx + c = 0 \), the product of the roots, \( \alpha \cdot \beta \), is determined using:

• \(\alpha \cdot \beta = \frac{c}{a}\)

Applying this to \( x^2 + 4x + 5 = 0 \), we substitute the given coefficients: \(\frac{5}{1} = 5\). Thus, the product of the roots is \(5\).

Understanding this information not only showcases deeper insights into the equation, but also spares you from solving the roots directly. These relationships are reliable and make working with quadratic equations much simpler.