Problem 17 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \((y+3)^{2}=0\)

Step-by-Step Solution

Answer

The solution to the equation \((y+3)^{2}=0\) is \(y = -3\)

Step 1: Inspect the given equation

Inspect the given equation \((y+3)^{2}=0\). It's clear that there's only one factor \((y+3)^{2}\) - which is squared. By the zero-product property, this factor must equal zero, so set \(y + 3 = 0\)

Step 2: Solve for 'y'

Solve for 'y' in equation \(y + 3 = 0\). Subtract 3 from both sides of the equation which leads to \(y = -3\) as the solution

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. These equations take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The goal is to find the values of \(x\) that satisfy the equation—or simply put—make the equation true. Quadratic equations are most easily solved using the quadratic formula, factoring, or by completing the square. Sometimes, like in the original problem, the zero-product property can make things simpler. By understanding these methods, students will have a toolkit for tackling various quadratic equations effectively and confidently. In our example, the equation is already quite simple, thanks to its form \( (y+3)^2 = 0 \), which can be directly addressed using the zero-product property.

Factoring

Factoring is a crucial part of solving algebraic equations, especially quadratics. It involves expressing a mathematical expression as a product of its factors, which are simpler expressions. For quadratic equations, the process typically aims to get them into a form where applying the zero-product property becomes straightforward.

To factor, one usually looks for two numbers that multiply to produce the constant term (\(c\)) and add to give the linear coefficient (\(b\)). However, sometimes, like in the case \((y+3)^2=0\), the equation is already factored. Recognizing this allows us to swiftly use the zero-product property to find solutions without additional steps. This technique saves time and simplifies problem-solving in algebra.

Algebraic Equations

Algebraic equations are mathematical statements that use algebraic expressions to demonstrate that two quantities are equal. They can range from simple linear equations to more complex quadratic or polynomial equations. Regardless of complexity, the main aim is to determine the values of the unknown variables that satisfy the equation. In solving algebraic equations, several methods can be applied, including substitution, elimination, and factoring.

• Linear equations, such as \(ax + b = 0\), involve only the first power of the variable.
• Quadratic equations like \(ax^2 + bx + c = 0\) involve the second power of the variable.

Each type requires a different approach for solving, emphasizing the flexibility and adaptability needed in algebra. Understanding these equations is key to mastering algebra, as they form the foundation for more advanced studies in mathematics.