Problem 63 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$(t+1) 2=2 t+7$$

Step-by-Step Solution

Answer

There is no solution.

Step 1: Simplify the Equation

Start by expanding and simplifying the equation. The given equation is \[(t+1)2 = 2t + 7\]First, distribute the \(2\) on the left side:\[2t + 2 = 2t + 7\]Notice that the \(2t\) terms on both sides cancel each other, leading to a simpler equation:\[2 = 7\]

Step 2: Check for No Solution or All Real Numbers

The simplified equation after cancelling \(2t\) from both sides was \[2 = 7\]This is a contradiction, as 2 can never equal 7. Hence, no values of \(t\) will satisfy this equation, implying there is no solution.

Key Concepts

Solving Equations

Solving equations is a fundamental algebraic skill that involves finding the values of variables that satisfy a given equation. Let's take a quadratic equation, for instance. The quadratic formula, \( ax^2 + bx + c = 0 \), is commonly used to find these solutions. For equations not in quadratic form, it's essential to first manipulate them into a standard structure. In our example equation, \((t+1)2 = 2t + 7\), the first step is to expand and simplify. We must distribute the 2 on the left-hand side, leading us to \(2t + 2 = 2t + 7\). The choice of simplification aids us in identifying the terms correctly and preparing the path for solving the equation. The goal is to isolate the variable (in this case, \(t\)) to one side, but in this situation, simplification reveals no solution, which will be discussed in the next section.

No Solution Case

Sometimes, during the process of solving an equation, you may encounter equations that have no solution. This is also known as an inconsistent equation. Such equations do not have any real numbers that satisfy them. In our example, after simplifying, we end up with \(2 = 7\). Clearly, this statement is a contradiction. No value of \(t\) can make this true, and hence, the equation has no solution.This situation arises when the same linear terms exist on both sides of the equation and cancel out, leading to a false statement. When you encounter a contradiction like this, it indicates that there is no possible value for the variable that will satisfy the original equation, meaning it's not solvable within the set of real numbers.

Step-by-Step Solution

Approaching equation problems step-by-step helps in systematically tackling complex problems. Let’s break down the steps used in our specific example:

• Step 1: Simplify the Equation - Distribute any constants and combine like terms to clean up the equation. In the example equation, \((t+1)2 = 2t + 7\), distributing gives \(2t + 2 = 2t + 7\). Cancel similar terms to simplify further.


• Step 2: Evaluate for No Solution - After simplifying, the left-hand side equals 2 while the right-hand side equals 7. Recognizing this tells us that the equation leads to a false statement (\(2 = 7\)), and thus there is no solution.

Following these structured steps can demystify the solving process. This methodical approach allows for identifying special cases like the no solution scenario, providing clarity on whether an equation can be solved meaningfully. By practicing this strategy, you can improve your ability to efficiently solve or simplify equations in the future.