Problem 6 - College Algebra Essentials

Question

Solve each equation in Exercises \(1-14\) by factoring. $$ 9 x^{2}+9 x+2=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1/3 and x = -2/3\).

Step 1: Rearrange equation

All terms are already on one side of the equation. Therefore, rearranging is not necessary in this context. We have \(9x^2 + 9x + 2 = 0\).

Step 2: Factor the quadratic equation

Factoring quadratics requires finding two numbers that multiply to give the product (in this case 2*9 = 18) and add up to give the coefficient of the linear term (in this case 9). However, 18 cannot be factored into two whole numbers that add up to 9. Therefore, we divide the equation by 9. The equation after division is \(x^2 + x + 2/9 = 0\). The numbers that fit this requirement are numbers 1/3 and 2/3 because (1/3)*(2/3) = 2/9 and 1/3 + 2/3 = 1. Therefore, the quadratic can be expressed as \((x + 1/3)(x + 2/3) = 0\).

Step 3: Solve for x

Having factored the equation, we can now use the zero product property to solve for x. The zero product property states that if a*b = 0, then either a = 0, b = 0, or both a and b are zero. In this case, we set each factor equal to 0 and solve for x, giving us \(x + 1/3 = 0 and x + 2/3 = 0\). Solving each gives \(x = -1/3 and x = -2/3\).

Key Concepts

Quadratic Equations

Quadratic equations represent expressions consisting of a squared variable, often fundamental in algebra. They take the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations are crucial because they appear in many mathematical problems and applications.

Understanding quadratic equations is key since they:

• Model a wide range of real-world situations.
• Help in predicting paths and determining optimal values.

Here, the given equation is \(9x^2 + 9x + 2 = 0\). Its significance lies in its ability to be factored, adding an extra level of simplification. Recognizing the structure of a quadratic equation is the first step. It assists in applying the appropriate methods, such as factoring, to find solutions efficiently.

Zero Product Property

The zero product property is a fundamental principle in algebra, especially when solving quadratic equations. It states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is pivotal in finding solutions to many algebraic equations.

Here's why it's important:

• Allows breaking down equations into simpler parts.
• Facilitates solving equations more intuitively by setting each factor to zero.

In the solved example, the equation \((x + 1/3)(x + 2/3) = 0\) uses this property. By setting each factor equal to zero, we find \(x = -1/3\) and \(x = -2/3\) as solutions. This step demonstrates how the zero product property transforms a complex expression into manageable pieces.

Solving Equations by Factoring

Factoring is a popular technique for solving quadratic equations. The aim is to express the quadratic equation in a product of two binomials. This method hinges on identifying terms that multiply to form the original quadratic in a simpler way.

The steps typically include:

• Identifying two numbers whose product and sum relate to the original coefficients.
• Writing the equation as a product of binomials.
• Applying the zero product property to solve for the variable efficiently.

In the example of \(9x^2 + 9x + 2 = 0\), we first attempted straightforward factoring but didn't find easy integers. Instead, the expression was simplified to \((x + 1/3)(x + 2/3) = 0\). This approach highlights the power and flexibility of factoring in making complex equations more tractable and solving them quickly.