Problem 38 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ y(y-1)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the variable y are y = 0 and y = 1.

Step 1: Apply the zero-factor property

To apply the zero-factor property, we need to set each factor equal to zero: $$y=0$$ and $$(y-1)=0$$.

Step 2: Solve the first equation

The first equation is already solved: $$y=0$$.

Step 3: Solve the second equation

To solve the second equation, add 1 to both sides of the equation: $$(y-1)+1=0+1$$ which simplifies to $$y=1$$.

Step 4: Present the solutions

The solutions to the given quadratic equation are $$y=0$$ and $$y=1$$.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). They involve terms up to the second degree, which means the highest exponent for the variable is 2. These equations are fundamental in algebra and appear in various real-life scenarios, such as physics and engineering. The standard form of a quadratic equation can be solved using several methods, including factoring, the quadratic formula, or completing the square. Quadratic equations can have different types of solutions, including:

• Two distinct real solutions
• One repeated real solution
• No real solution but two complex solutions

Finding these solutions often requires a clear understanding of algebraic principles and sometimes involves looking at the discriminant \( b^2 - 4ac \), which helps us determine the nature of the roots.

Solving Equations

Solving equations is the process of finding the value(s) of the variable(s) that make the equation true. In the context of quadratic equations, solving usually involves determining the values of the variable that satisfy the equation. One of the simplest ways to solve a quadratic equation is by using the zero-factor property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property becomes very useful when an equation is factored properly.Here's a step-by-step approach:

• Transform the equation into a factored form like \( (x - p)(x - q) = 0 \)
• Apply the zero-factor property, setting each factor equal to zero (\( x - p = 0 \) and \( x - q = 0 \))
• Solve these simpler linear equations to find the values of \( x \)

The zero-factor property simplifies solving quadratic equations significantly since it reduces a potentially complex problem into simpler linear equations to solve.

Factorization

Factorization in algebra involves breaking down a complex expression into a product of simpler factors. For quadratic equations, often the first step is to express the quadratic expression in a factored form. This process is called factorizing. When a quadratic is written as a product of two binomials, it can be easily set to zero and solved using the zero-factor property. Factorizing a quadratic equation typically involves:

• Identifying two numbers that multiply to give \( ac \) (product of coefficients \( a \) and \( c \)) and add to \( b \) (the coefficient of the linear term)
• Rewriting the quadratic equation using these two numbers
• Grouping terms and factoring each group
• Factoring the entire expression to find the simplified factors

Understanding factorization is key to simplifying and solving quadratic equations efficiently. It is often the first step before applying further algebraic methods like the zero-factor property.