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Chapter 5: Problem 15

Solve. $$(2 t+5)(t-7)=0$$

Short Answer

Expert verified
The solutions are \(t = -\frac{5}{2}\) and \(t = 7\).

Step by step solution

01

- Identify factors

Observe that the given equation is already factored into two binomials: \((2t + 5)(t - 7) = 0\).
02

- Apply Zero Product Property

Using the Zero Product Property, set each factor equal to zero: \(2t + 5 = 0\) and \(t - 7 = 0\).
03

- Solve for t in the first equation

Solve the first equation, \(2t + 5 = 0\): Subtract 5 from both sides to get \(2t = -5\). Then divide both sides by 2: \(t = -\frac{5}{2}\).
04

- Solve for t in the second equation

Solve the second equation, \(t - 7 = 0\): Add 7 to both sides to get \(t = 7\).
05

- State the solutions

The solutions to the equation \((2t + 5)(t - 7) = 0\) are \(t = -\frac{5}{2}\) and \(t = 7\).

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Key Concepts

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

Factoring Binomials
Factoring binomials involves expressing an expression as the product of two or more simpler expressions. Here, the equation \((2t + 5)(t - 7) = 0\) is already factored into two binomials: \((2t + 5)\) and \((t - 7)\).
  • The equation \((2t + 5)\) means that we need to find the value of \ t \ which makes \ 2t + 5 \ equal to zero.
  • Similarly, \(t - 7\) means we need to find the value of \ t \ which makes \ t - 7 \ equal to zero.
Understanding how to factor binomials is crucial because it simplifies many algebraic problems, making them easier to solve.
Solving Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). In our example, the equation is factored as \((2t + 5)(t - 7) = 0\). To solve these equations, we can use the Zero Product Property, which states that if the product of two factors is zero, at least one of the factors must be zero.
Using this property:
  • First, set \(2t + 5\) equal to 0 and solve for \ t \: \(2t + 5 = 0\Rightarrow 2t = -5 \Rightarrow t = -\frac{5}{2}\).
  • Next, set \(t - 7\) equal to 0 and solve for \ t \: \(t - 7 = 0\Rightarrow t = 7\).
Solving quadratic equations usually involves finding two values of the variable that make the equation true.
Linear Equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In the context of our example, we solve two linear equations derived from the factored form:
  • The first linear equation is \2t + 5 = 0\. To isolate \ t \, subtract 5 from both sides: \2t = -5\. Then, divide both sides by 2: \ t = -\frac{5}{2}\.
  • The second linear equation is \(t - 7 = 0\). To solve for \ t \, add 7 to both sides: \ t = 7\.
These solutions show the values of \ t \ that satisfy the original equation. Linear equations are fundamental in algebra, and solving them requires straightforward techniques like addition, subtraction, multiplication, and division.

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