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Chapter 7: Problem 331

Solve. \(7 a^{2}+14 a=7 a\)

Short Answer

Expert verified
The solutions are \(a = 0\) and \(a = -1\).

Step by step solution

01

Move All Terms to One Side

Start by moving all terms to one side of the equation to set it to zero. Subtract \(7a\) from both sides: \[7a^2 + 14a - 7a = 0\]Simplify the terms: \[7a^2 + 7a = 0\]
02

Factor Out the Greatest Common Factor

Notice that both terms on the left have a common factor of \(7a\). Factor \(7a\) out of the equation: \[7a(a + 1) = 0\]
03

Apply the Zero Product Property

Set each factor equal to zero. The zero product property states that if a product of factors is zero, then at least one of the factors must be zero. Set the first factor equal to zero: \[7a = 0\]Then set the second factor equal to zero: \[a + 1 = 0\]
04

Solve for \(a\)

Solve each equation for \(a\). For the first equation: \[7a = 0 \rightarrow a = 0\]For the second equation: \[a + 1 = 0 \rightarrow a = -1\]

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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 where we break down an expression into a product of simpler factors. This is an essential skill for solving quadratic equations. In our problem, the equation was simplified to: 7a^2 + 14a - 7a = 0 which we then simplified to: 7a^2 + 7a = 0. The next step was factoring out the greatest common factor (GCF). Recognizing that both terms share a common factor of 7a, we factored it out: 7a(a + 1) = 0. Factoring allows us to break down the equation into simpler components, making it easier to solve.
Zero Product Property
The Zero Product Property is a principle that states if the product of two factors is zero, then at least one of the factors must be zero. In our factored equation: 7a(a + 1) = 0, we apply this property by setting each factor to zero. Solving the equations: 7a = 0 and a + 1 = 0, we find the solutions a = 0 and a = -1. The Zero Product Property ensures that we explore all possible solutions for the variable.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor shared by terms in an equation. Finding the GCF helps simplify the equation, making it easier to work with. In our example, the terms 7a^2 and 7a share a GCF of 7a. By factoring out 7a, we simplify the original equation 7a^2 + 7a = 0 to the product 7a(a + 1) = 0. This step is crucial in solving the equation because it transforms it into a form where we can apply the Zero Product Property to find the solutions for a.

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