Problem 60 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ (2 y+5)(2 y+5)=0 $$

Step-by-Step Solution

Answer

Answer: The value of y is -5/2 or -2.5.

Step 1: Understand the zero-factor property

The zero-factor property states that if two or more factors are multiplied together and the product is zero, then at least one of those factors must be zero. In this case, the equation is of the form (2y + 5)(2y + 5) = 0, so we need to set each factor equal to zero and solve for y.

Step 2: Set each factor equal to zero

We will set each factor equal to zero. As both factors are the same, we just need to solve for one of them: $$ 2y + 5 = 0 $$

Step 3: Solve for y

Now, we will isolate y: $$ 2y = -5 \\ y = -\frac{5}{2} $$ So, the solution for the equation is: $$ y = -\frac{5}{2} $$

Key Concepts

Solving Quadratic Equations

Quadratic equations are a fundamental part of algebra and come in the form of \( ax^2 + bx + c = 0 \). These equations can often be solved using different techniques such as factoring, completing the square, or using the quadratic formula.

The zero-product property plays a crucial role when solving quadratic equations, particularly when they can be factored into binomials. It allows us to find solutions or roots of the equation by setting each factor equal to zero.

• Identify the equation and confirm it is quadratic.
• Attempt to factor the equation if possible.
• Use zero-product property to set each factor equal to zero and solve for the variable.

It's important to verify solutions by substituting back into the original equation to ensure correctness. This process helps solidify understanding and reinforces the steps in solving quadratic equations effectively.

Factoring

Factoring is breaking down equations into simpler expressions or 'factors' that can be multiplied to give the original equation. In the exercise provided, the equation \((2y + 5)(2y + 5) = 0\) is already factored. When equations are not immediately in factorable form, you may first need to apply systematic techniques to break them down.

• Look for common factors in all terms.
• Apply different methods like grouping, using special product forms (e.g., difference of squares), or trial and error.

Once factored, each component of the factored expression can be solved individually using the zero-product property. This method simplifies complex expressions and makes finding solutions more accessible.

Algebraic Equations

Algebraic equations involve finding the value of unknowns that satisfy given mathematical expressions. They use numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

The equation provided in the example, \((2y + 5)(2y + 5) = 0\), is an algebraic equation where the power of the variable \(y\) is 2, classifying it as quadratic.

• Equations can have different forms: linear, quadratic, cubic, etc.
• A deep understanding of the types of equations helps in choosing the correct solution method.
• These equations are foundational for various mathematical concepts and real-life problem solving.

Tackling algebraic equations involves systematically manipulating and applying algebraic principles like the zero-product property to arrive at the solution.