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Chapter 6: Problem 4

Use the Zero-Factor Property to solve the equation. $$ (t-3)(t+8)=0 $$

Short Answer

Expert verified
The solutions of the equation \( (t-3)(t+8)=0 \) are \( t=3 \) and \( t=-8 \).

Step by step solution

01

Apply the Zero-Factor Property

According to the Zero-Factor Property, if a product of two numbers equals zero, then at least one of the factors must be equal to zero. Therefore, in the given equation \( (t-3)(t+8) = 0 \), we can say that either \( (t-3) = 0 \) or \( (t+8) = 0 \) or both.
02

Solve for t in First Equation

Set the first factor, \( (t-3) \), equal to zero and solve for t: \( t-3 = 0 \) Add 3 to both sides of the equation, this gives: \( t = 3 \) So, the first solution for the equation is 3.
03

Solve for t in Second Equation

Next, set the second factor, \( (t+8) \), equal to zero and solve for t: \( t+8 = 0 \) Subtract 8 from both sides of the equation, this gives: \( t = -8 \) So, the second solution for the equation is -8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a mathematical process used to break down expressions into products of simpler expressions. This is especially useful for solving equations, as it can simplify complex expressions into more manageable forms. The equation given,
\((t-3)(t+8)=0\),
is already factored into two binomial expressions. Here, the expression is in the simplest factorized form before applying the Zero-Factor Property, which is crucial for finding solutions.
  • Factoring involves finding numbers or expressions that multiply together to give the original expression.
  • The goal is to express the parent equation in such a way that makes it easy to identify zero points, or roots.
  • For quadratic equations, factoring is an effective method when they can be easily rewritten as a product of binomials.
Factoring helps to simplify the process of finding solutions to equations by organizing them into smaller, solvable parts.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that make the equation true. In the context of our exercise, the Zero-Factor Property plays a key role. This property makes solving factored equations straightforward. When you have a factored quadratic equation set to zero, you can use each factor to find solutions.
  • Start by ensuring the quadratic equation is set to zero. This is necessary to apply the Zero-Factor Property.
  • The Zero-Factor Property states that if \[ a \cdot b = 0 \], then either \( a = 0 \) or \( b = 0 \) or both.
  • Apply this property separately to each factor of the equation, just like in our exercise with \( (t-3)=0 \) and \( (t+8)=0 \).
By solving these simpler equations, we identify possible values for the variable.
Roots of Equations
Roots of an equation are solutions that make the equation true. For a quadratic equation such as \((t-3)(t+8)=0\), these roots are the specific values of \( t \) that satisfy the equation.
  • When you factor a quadratic equation and apply the Zero-Factor Property, you are essentially finding its roots.
  • In this case, the equation has two roots: \( t=3 \) and \( t=-8 \), determined by solving \((t-3)=0\) and \((t+8)=0\).
  • The roots of the equation represent the points where the graph of the equation would intersect the x-axis, if you plot it out.
Understanding roots helps in visualizing and interpreting the solutions to quadratic equations, giving you a clearer insight into the behavior of the graph of an equation.

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