Problem 27 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \((a-20)(a+15)=0\)

Step-by-Step Solution

Answer

The solution comprises of two possible values for the variable 'a', hence \(a = 20\) and \(a = -15\).

Step 1: Identify the Components of the Equation

The equation has two factors, \((a-20)\) and \((a+15)\), equated to zero.

Step 2: Apply the Zero-Product Property

The zero-product property allows us to set each factor equal to zero. Therefore, we get two sub-equations: \((a-20) = 0\) and \((a+15) = 0\)

Step 3: Solve Each Sub-Equation

Solving for 'a' in each of the equations gives us the possible values of 'a'. For equation \((a-20) = 0\), add 20 to both sides to obtain \(a = 20\). Similarly for equation \((a+15) = 0\), subtract 15 from both sides to obtain \(a = -15\)

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, we use letters like \(a\) or \(b\) to represent numbers or quantities that are not yet known. This allows us to create equations and formulas that describe mathematical relationships.

There are various operations in algebra, such as addition, subtraction, multiplication, and division, which are used to solve equations. By following these operations, we can find the values of the variables that satisfy a given equation. The use of algebra is foundational in problem solving and helps us understand patterns and structures in mathematics. It also forms the basis for more advanced topics in mathematics like calculus and geometry.

Equation Solving

Equation solving is about finding the value of the variable that makes an equation true. To solve an equation means to determine what number, when substituted for the variable, will make both sides of the equation equal. Let's take a look at solving this particular equation:

• First, identify the components of the equation, which in this case are \((a-20)(a+15)=0\). We see that it's expressed as a product of two factors.
• To solve the equation, we apply the zero-product property. This property is fundamental because if the product of two factors is zero, then at least one of the factors must be zero. Thus, we split the equation into two separate equations: \(a-20=0\) and \(a+15=0\).
• Next, solve each equation separately. For \(a-20=0\), add 20 to both sides, resulting in \(a=20\). For \(a+15=0\), subtract 15 from both sides, resulting in \(a=-15\).

Solving equations requires careful manipulation of the variables and constant terms on both sides. Through systematic steps, you find the values that satisfy the original equation.

Factoring

Factoring is the process of breaking down an expression into a product of its simplest components, or factors. It is an essential skill in algebra that simplifies equations, making them easier to solve. Factoring is particularly useful when dealing with polynomial equations, like the one seen in the original exercise, \((a-20)(a+15)=0\).

Here are some key points about factoring:

• It involves finding numbers or expressions that multiply together to give the original expression. For example, the expression \((a-20)(a+15)\) is already factored into two linear factors.
• Different techniques of factoring include finding common factors, using special patterns like the difference of squares, and grouping.
• Factoring can reveal solutions to equations based on the zero-product property, which is applied by setting each factor to zero.

Understanding how to factor correctly allows you to simplify complex equations, making it easier to find solutions and understand the underlying mathematical relationships.