Problem 14 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

Sum is -2, product is 0.

Step 1: Equation Rearrangement

The given equation is already in a simplified quadratic form: \[x^2 + 2x = 0\] It is necessary to recognize this as a standard quadratic equation where \(a = 1\), \(b = 2\), and \(c = 0\).

Step 2: Identify the Relationships

In a quadratic equation \(ax^2 + bx + c = 0\), sum of the roots \(\alpha + \beta = -\frac{b}{a}\), and product of the roots \(\alpha \cdot \beta = \frac{c}{a}\).

Step 3: Calculate the Sum of Roots

Applying \(\alpha + \beta = -\frac{b}{a}\), we substitute \(b = 2\) and \(a = 1\):\[\alpha + \beta = -\frac{2}{1} = -2\]

Step 4: Calculate the Product of Roots

Using \(\alpha \cdot \beta = \frac{c}{a}\), substitute \(c = 0\) and \(a = 1\) into the formula:\[\alpha \cdot \beta = \frac{0}{1} = 0\]

Key Concepts

Sum of Roots

The sum of roots of a quadratic equation is an interesting property. It comes from the coefficients of the equation in its standard form. Suppose you have a quadratic equation in standard form:

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are constants. The sum of the roots, often represented as \(\alpha + \beta\), can be found using the formula:

• \(\alpha + \beta = -\frac{b}{a}\)

This formula shows that the sum of roots is simply the negation of the coefficient of \(x\) (which is \(b\)) divided by the coefficient of \(x^2\) (which is \(a\)). It is derived from adding the roots obtained through solving the quadratic by factoring or using the quadratic formula.
The formula remains consistent no matter the actual values of the roots. This provides an efficient way to determine the sum without needing to solve the equation fully. For instance, in the quadratic equation \(x^2 + 2x = 0\), we found that \(b = 2\) and \(a = 1\). Therefore, the sum of the roots is \(-\frac{2}{1} = -2\).
Such shortcuts are not only nifty but also immensely helpful under test conditions or when quick analyses are required.

Product of Roots

Calculating the product of the roots of a quadratic equation provides another useful insight just like the sum. The product of the roots \(\alpha \cdot \beta\) is obtained from the constant term within the equation's standard form:

• \(ax^2 + bx + c = 0\)

The formula to find the product of the roots is:

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

This means that the product of the roots of a quadratic equation can be directly found by dividing the constant term \(c\) by the coefficient of the quadratic term \(a\).
For example, in the equation \(x^2 + 2x = 0\), where \(a = 1\) and \(c = 0\), the product of the roots evaluates to \(\frac{0}{1} = 0\). It showcases the ease of finding products without solving for the roots themselves.
Understanding and employing these relationships can be particularly beneficial when evaluating the nature of the roots or inferring additional properties, such as the relationship between the roots in certain types of equations.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is foundational for accessing its properties easily. This form standardizes the structure and positions of the quadratic equation like this:

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are known as coefficients, where \(a\) cannot be zero as it would otherwise not remain a quadratic equation.
Understanding the standard form helps in recognizing the equation and directing various methods for predictions without fully solving it, such as calculating the sum or product of roots.
For the equation \(x^2 + 2x = 0\), recognizing it in the form \(ax^2 + bx + c = 0\) allowed us to identify \(a = 1\), \(b = 2\), and \(c = 0\), directly owing to the standard form.
Utilizing the standard form not only helps in solving but also aids in graphing and analyzing the behavior of the quadratic function, predicting its shape based on the signs and values of the coefficients.