Problem 30 - Algebra Form and Function

Question

Without solving the equation, decide how many solutions it has. $$ (x-1)(x-2)=0 $$

Step-by-Step Solution

Answer

Answer: The equation has 2 solutions.

Step 1: Identify the Factors

In the given equation \((x-1)(x-2)=0\), there are two factors: \((x-1)\) and \((x-2)\).

Step 2: Consider the Zero Product Property

The Zero Product Property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have \((x-1)(x-2)=0\). So, either \((x-1)=0\) or \((x-2)=0\).

Step 3: Count the Distinct Solutions

Since there are two distinct factors, we have two possible solutions for \(x\). So, the given equation has 2 solutions.

Key Concepts

Zero Product Property

The zero product property is a fundamental concept in algebra. It states that if the product of two or more factors equals zero, then at least one of these factors must be zero. This property forms the basis for solving quadratic equations by factoring.
In an equation like

• \((a)(b) = 0\)

we can conclude that either

• \(a = 0\)
• or \(b = 0\)

This is because multiplying any number by zero results in zero. Understanding this property helps in identifying solutions to equations quickly.

Factoring

Factoring involves expressing a polynomial as the product of its simplest components or factors. For quadratic equations, this often means breaking down the expression into binomials.
Take the example:

• \((x-1)(x-2) = 0\)

This expression is already factored, showcasing two binomials: \((x-1)\) and \((x-2)\).
This step prepares the equation for applying the zero product property. Factoring transforms complex expressions into simpler forms, making it easier to find solutions.

Solutions of Equations

Solutions of an equation are the values that satisfy the equation, making it true. For the equation

• \((x-1)(x-2) = 0\),

you determine solutions using the zero product property. Here, the solutions are where each factor equals zero:

• When \((x-1) = 0\), solving gives \(x = 1\).
• When \((x-2) = 0\), solving gives \(x = 2\).

Thus, the equation has two solutions: \(x = 1\) and \(x = 2\). Each solution corresponds to one factor. This method shows how factoring and the zero product property help find the exact solutions of quadratic equations.