Problem 21 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ -7 a(3 a+10)=0 $$

Step-by-Step Solution

Answer

The solutions are \(a = 0\) and \(a = -\frac{10}{3}\).

Step 1: Understand the Equation

We have the equation \[-7a(3a+10)=0\].This is a product of two expressions equaling zero: \[-7a\] and \[(3a+10)\]. According to the Zero Product Property, if a product of two factors is zero, then at least one of the factors must be zero.

Step 2: Apply the Zero Product Property

Using the Zero Product Property, we set each factor equal to zero:1. \(-7a = 0\)2. \(3a + 10 = 0\)

Step 3: Solve First Equation \(-7a = 0\)

To solve the equation \(-7a = 0\), divide both sides by -7:\[\frac{-7a}{-7} = \frac{0}{-7} \a = 0\]So the first solution is \(a = 0\).

Step 4: Solve Second Equation \(3a + 10 = 0\)

To solve \(3a + 10 = 0\), first subtract 10 from both sides:\[3a = -10\]Then, divide both sides by 3:\[a = -\frac{10}{3}\]So the second solution is \(a = -\frac{10}{3}\).

Step 5: Verify Solutions

We've determined that the solutions to the equation are \(a = 0\) and \(a = -\frac{10}{3}\). Substituting these values back into the original equation verifies that both make the equation true.

Key Concepts

Quadratic Equations

Quadratic Equations are a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). These equations are called quadratic because "quad" means square, indicating the highest power of the variable is 2.
Quadratic equations are essential in algebra as they help us model various real-life situations and predict outcomes.In the given exercise, the equation \( -7a(3a + 10) = 0 \), once expanded, could be represented as a standard quadratic equation. Here, it is in factored form, making it easier to solve using the Zero Product Property.
To solve a quadratic equation, we can utilize various methods like factoring, using the quadratic formula, or completing the square. Understanding these methods broadens our ability to tackle quadratic challenges in different contexts.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In the context of quadratic equations like in the exercise, solving involves setting each factor equal to zero based on the Zero Product Property.
This equation-solving process is step-by-step:

• Step 1: Identify and understand the equation.
• Step 2: Use relevant mathematical properties, like the Zero Product Property, to simplify the equation.
• Step 3: Solve for the variable, one part of the equation at a time.
• Step 4: Verify your solution by substituting it back into the original equation.

The Zero Product Property is especially helpful in situations like in our example, where the equation is presented in a product form. This property states that if the product of two numbers is zero, then at least one of the numbers must be zero.
Math-based problem solving is an iterative process that involves these strategic, logical steps to reach a solution.

Algebraic Expressions

Algebraic Expressions are mathematical phrases that include numbers, variables, and operation symbols. They are the building blocks of equations and can have one or more terms. In the equation \( -7a(3a + 10) = 0 \), the two distinct expressions are \(-7a\) and \(3a + 10\), linked together by multiplication.Understanding how to manipulate algebraic expressions is crucial when solving equations. We apply various rules and properties, like distribution or factorization, to simplify and solve for unknown variables.
Key points in handling algebraic expressions include:

• Recognizing like terms that can add or subtract.
• Applying multiplication over addition using the distributive property, if needed.
• Transitioning between factored and expanded forms of expressions for ease of solving.

Mastering algebraic expressions equips you with the flexibility to rewrite equations and reveal solutions that might be otherwise hard to see. It is a powerful tool for simplifying the path from problem to solution.