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Chapter 3: Problem 84

For Problems \(69-84\), solve each of the equations. $$ -5 a=-a^{2} $$

Short Answer

Expert verified
The solutions are \(a = 0\) and \(a = 5\).

Step by step solution

01

Rearrange the equation

The given equation is \(-5a = -a^2\). Begin by rearranging all terms to one side of the equation to set it equal to zero. Add \(a^2\) to both sides: \(a^2 - 5a = 0\).
02

Factor the equation

Factor the expression on the left side of the equation. Find the greatest common factor of the terms: \(a(a - 5) = 0\).
03

Solve each factor

Set each factor equal to zero and solve for \(a\). For \(a = 0\), the solution is \(a = 0\). For \(a - 5 = 0\), solve to get \(a = 5\).
04

Compile the solutions

The solutions to the equation are the values of \(a\) that satisfy both factors. Hence, the solutions are \(a = 0\) and \(a = 5\).

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Key Concepts

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

Factoring
Factoring is a crucial step in solving quadratic equations, including those of the form seen in our exercise. When we factor, we are essentially breaking down a complex expression into simpler components, or "factors," that multiply to give the original expression. This is similar to finding the prime factors of a number, but here, it鈥檚 for algebraic expressions. In the equation \(a^2 - 5a = 0\), our goal is to factor out the greatest common factor (GCF) from the terms.
  • The terms \(a^2\) and \(-5a\) share a common factor, which is \(a\).
  • By factoring out \(a\), the expression becomes \(a(a - 5) = 0\).
Factoring simplifies the equation, allowing us to easily apply the zero-product property to find solutions. Once factored, each component can be set to zero, thus simplifying the process of solving the equation.
Solving Equations
Solving equations involves finding the values of the variables that make the equation true. In our quadratic equation case, the solution process involves using the factored form \(a(a - 5) = 0\). The zero-product property is a fundamental principle used here, stating that if a product of two factors equals zero, at least one of the factors must be zero.
  • First, set each factor equal to zero: \(a = 0\) and \(a - 5 = 0\).
  • Solving \(a = 0\) gives us a direct solution: \(a = 0\).
  • For \(a - 5 = 0\), add 5 to both sides to isolate \(a\), yielding \(a = 5\).
Through this method, we determine all possible solutions that satisfy the equation. Thus, we find that the zeros of the quadratic equation \(-5a = -a^2\) are \(a = 0\) and \(a = 5\).
Algebraic Expressions
Algebraic expressions are like the building blocks of equations. They consist of numbers, variables, and operations that define a mathematical phrase. In our problem, the expression \(-5a = -a^2\) is manipulated to solve the equation using algebraic principles. When dealing with such expressions:
  • Combine like terms: Start by rearranging terms to one side to simplify further actions, seen here as bringing all terms to equal zero.
  • Simplification: Reducing expressions helps make the factorization process easier, as done when transforming the equation to \(a^2 - 5a = 0\).
  • Utilizing properties: Recognizing when to apply properties like the zero-product property aids in efficient problem-solving.
Understanding how to handle and manipulate algebraic expressions is key to mastering the solving of quadratic equations and plays a significant role in higher-level algebra concepts.

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 3 x^{2}=75 $$

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ 2 x y+6 x+y+3 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 6 x^{2}+25 x+14=0 $$

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ x^{4}-5 x^{2}-36 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 8 n^{2}-6 n-5=0 $$
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