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Chapter 2: Problem 94

Solve each equation. $$ t^{2}-6 t=t(8+t) $$

Short Answer

Expert verified
The solution to the equation is \(t = 0\).

Step by step solution

01

Expand the Equation

Begin by expanding the right side of the equation: \[t(8+t) = 8t + t^2\]Substitute this into the equation to get: \[t^2 - 6t = 8t + t^2\]
02

Simplify the Equation

Subtract \(t^2\) from both sides to eliminate \(t^2\): \[t^2 - t^2 - 6t = 8t + t^2 - t^2\]This simplifies to:\[-6t = 8t\]
03

Rearrange the Equation

Add \(6t\) to both sides to isolate \(t\):\[-6t + 6t = 8t + 6t\]This simplifies to:\[0 = 14t\]
04

Solve for t

Divide both sides by \(14\) to solve for \(t\):\[\frac{0}{14} = \frac{14t}{14}\]Simplifying gives \(t = 0\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In the given equation, \[t^2 - 6t = t(8 + t)\].We have expressions on both sides:
  • Left-hand side: \(t^2 - 6t\), which consists of a quadratic term \(t^2\) and a linear term \(-6t\).
  • Right-hand side: \(t(8 + t)\), which is a product of \(t\) and the expression \((8 + t)\).

To solve such problems, our first task is to manipulate these expressions appropriately, making sure we follow the correct math rules. This helps us transform the equation into a form that is easier to solve. Algebraic expressions are the building blocks in algebra, allowing us to form and solve equations.
Solving Equations
Solving equations involves finding the value(s) of the variable(s) that make the equation true. In this guide, we will solve for the variable \(t\) in the equation:\[t^2 - 6t = t(8 + t)\]
Our goal is to find all possible values of \(t\) that satisfy this equation.
  • First, expand the right-hand side to get \(8t + t^2\).
  • Next, simplify the equation by combining like terms and eliminating excess variables, leading to \(-6t = 8t\).
  • Rearrange the equation to isolate the term with \(t\), obtaining the simple linear equation \(0 = 14t\).
By solving this linear equation, we determine that \(t = 0\) is the solution. This process involves utilizing algebraic properties to transform the original equation into simpler, more manageable forms.
Algebraic Manipulation
Algebraic manipulation involves techniques to restructure and simplify expressions or equations. In solving \(t^2 - 6t = t(8 + t)\), we apply several key manipulations:
  • Distributive Property: Use it to expand \(t(8 + t)\) into \(8t + t^2\).
  • Combining Like Terms: After expansion, compare both sides of the equation, aligning terms with the same degree.
  • Elimination: Remove identical terms from each side by subtracting or adding them, aiming to simplify the equation further.
  • Rearrangement: Move terms to isolate variables on one side of the equation by using basic arithmetic operations.

These steps guide us in refining the equation step-by-step. Practicing these manipulations improves proficiency in solving more complex equations and deepens understanding of algebra's fundamental concepts.

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