Problem 45 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ n(n-10)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions to the given equation are \(n = 0\) and \(n = 10\).

Step 1: Apply the zero-factor property

Since we have two factors \(n\) and \((n-10)\), by applying the zero-factor property, we can set each factor equal to zero: $$ n = 0 \\ (n - 10) = 0 $$

Step 2: Solve for the variable n

For the first factor, \(n=0\), we already have the solution. It's \(n=0\) To solve for the second factor \((n-10)=0\), we need to add 10 to both sides of the equation to isolate \(n\): $$ n - 10 + 10 = 0 + 10 \\ n = 10 $$ So the second solution is \(n=10\)

Step 3: State the solutions

The zero-factor property allowed us to find the two solutions for the given equation: \(n = 0\) and \(n = 10\).

Key Concepts

Quadratic Equations

Quadratic equations form the foundation of many algebraic concepts. These equations take the expression of either \[ax^2 + bx + c = 0 \]or a simplified factored form, like the equation in our problem: \[n(n-10) = 0\].
Quadratic equations are recognizable by the term \(x^2\), known as the quadratic term. The letter "a" determines the equation's form, while "b" is the coefficient of \(x\), and "c" is the constant term.
The equation we are working with simplifies this idea by using a format that showcases its factorized state directly. Such setups help demonstrate the solution methods with clarity and efficiency. Understanding this makes solving equations like these more intuitive and straightforward for students.

Factoring

Factoring is a crucial algebraic technique that involves expressing a polynomial as a product of its factors. This process transforms the original quadratic equation into something more manageable and often simpler to solve.
For instance, in our problem, the equation \(n(n-10) = 0\) has already been factored into two components: \(n\) and \(n-10\). Factoring in algebra can involve several techniques:

• Simple Factoring: Useful when dealing with basic equations like our current one encompassed in a simpler format.
• Grouping and Special Products: More complex polynomials might need grouping terms or recognizing patterns (like difference of squares).

By factoring an equation, we can directly apply further mathematical tools, such as the zero-factor property, simplifying our path to finding solutions. Mastering factoring is key to progressing within algebra.

Algebraic Solutions

Algebraic solutions are the end result of unraveling equations, using various techniques like factoring, to find values that satisfy the equation. By searching for solutions algebraically, one learns to manipulate the equation to reveal its hidden solutions.
In this specific context, the zero-factor property acts as our primary tool for simplifying and solving. Once an equation is factored to \[n(n-10) = 0\], \[each factor can be individually set to zero:\]

• If \(n = 0\), then \(n\) is solved directly, giving one solution: \(n = 0\).
• For \((n-10) = 0\), solving involves basic algebra, ultimately isolating \(n\) to find \(n = 10\).

This method demonstrates how algebra simplifies complexities using straightforward logic. These solutions show how factoring and properties like the zero-factor property make it possible to untangle quadratic equations efficiently.