Problem 42 - Basic Technical Mathematics with Calculus

Question

$$ \text { Solve the given quadratic equations by factoring.}$$ In analyzing the path of a rocket, the equation \(16 t^{2}=320 t\) is used. Solve for \(t\)

Step-by-Step Solution

Answer

The solutions are \(t = 0\) and \(t = 20\).

Step 1: Rearrange the Equation

Rearrange the equation so that all terms are on one side of the equation, setting the equation to zero. \[ 16t^2 - 320t = 0 \]

Step 2: Factor Out the Greatest Common Factor

Identify and factor out the greatest common factor (GCF) from the terms. The GCF of the terms is \(16t\). Factoring this out, we get: \[ 16t(t - 20) = 0 \]

Step 3: Apply the Zero Product Property

Apply the zero product property, which states that if a product of factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: \[ 16t = 0 \] or \[ t - 20 = 0 \]

Step 4: Solve Each Equation

Solve each equation from the previous step. For the first equation, \(16t = 0\), solve for \(t\) by dividing both sides by 16: \[ t = 0 \] For the second equation, \(t - 20 = 0\), solve for \(t\) by adding 20 to both sides: \[ t = 20 \]

Step 5: List the Solutions

The solutions to the original equation are \(t = 0\) and \(t = 20\). These are the times when the path of the rocket reaches the specified condition.

Key Concepts

Factoring

Factoring is a core technique used in solving quadratic equations. When we factor an expression, we look for numbers or variables that can multiply together to recreate the original equation. In simpler terms, factoring breaks down a complex mathematical expression into simpler, smaller parts.

• In our exercise, we started with the equation \[ 16t^2 - 320t = 0 \]
• Our goal was to simplify it by factoring.
• The simplest form of the equation often makes solving the quadratic much easier.

By identifying common factors and using them to rewrite the equation, we can find solutions more easily.

Zero Product Property

The zero product property is an essential principle in algebra. It states that if the product of any set of numbers is zero, then at least one of the numbers must be zero. This is very important when dealing with factored equations.Consider our equation after factoring: \[16t(t - 20) = 0\]

• The zero product property tells us that either \(16t = 0\) or \(t - 20 = 0\).
• That means we can set each factor to zero to find the possible solutions for \(t\).

This property is the cornerstone for solving quadratic equations through factoring, as it allows us to break down complex equations into manageable parts.

Greatest Common Factor

The greatest common factor, or GCF, is the largest factor that divides two or more numbers. Identifying the GCF is the first step in the factoring process. This step simplifies the original equation and sets up subsequent operations.In the exercise, we determined that the GCF of \[16t^2 - 320t\]was \(16t\). This allowed us to factor the expression to \[16t(t - 20) = 0\].

• Finding the GCF helps in simplifying equations and reducing calculation effort.
• The GCF will always be comprised of numerical and variable components that are common across terms.

Understanding how and why to find the GCF simplifies the path to finding solutions to quadratic equations.

Solving Equations

Solving quadratic equations involves finding the value of the variable that makes the equation true. This involves using several techniques like factoring and the zero product property.After factoring the equation \(16t(t - 20) = 0\)we applied the zero product property. This resulted in two simpler equations:

• \(16t = 0\)
• \(t - 20 = 0\)

Solving these produces the time values \(t = 0\) and \(t = 20\).

• Solving equations often involves manipulating them by adding, subtracting, multiplying, or dividing to isolate the variable.
• Both solutions must fit within the context of the original problem, such as the scenario of a rocket's path.

Understanding the complete process ensures you can tackle any quadratic problem with confidence.