Problem 65 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 5(x-9)(x-6)=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(5(x-9)(x-6)=0\) are \(x=9\) and \(x=6\).

Step 1: Analyzing the Zero Product Property

The equation \(5(x-9)(x-6)=0\) features a product of three factors: 5, \((x-9)\), and \((x-6)\). Using the Zero-Product Property, these are set individually equal to zero in order to solve for the variable \(x\). Because 5 is a constant, it contributes no solutions. The equalities to be solved are therefore:\[(x-9)=0 \quad (1)\]\[(x-6)=0 \quad (2)\]

Step 2: Solving Equality (1)

Equality (1) simplifies to:\[x-9=0\]Solving this equation means isolating \(x\) by adding 9 to both sides:\[x=9\]

Step 3: Solving Equality (2)

Equality (2) simplifies to:\[x-6=0\]Solving this equation means isolating \(x\) by adding 6 to both sides:\[x=6\]

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is a foundational skill in algebra. A quadratic equation typically takes the form \( ax^2+bx+c=0 \), where \( a \), \( b \), and \( c \), are constants, and \( x \) represents the variable or unknown.

When working with quadratic equations, various methods can be used, including factoring, using the quadratic formula, completing the square, or graphing. Factoring is a common method when the equation can easily be broken down into the product of binomials or monomials. Such a scenario is presented by the equation in our exercise:\[ 5(x-9)(x-6)=0 \], which shows the equation already factored.

The Zero Product Property comes into play here, asserting that if the product of several numbers is zero, at least one of the numbers must be zero. Applying this property, we break the equation into simpler parts, each set equal to zero to find the values of \( x \) that fulfill the original equation.

Factoring Polynomials

Factoring polynomials is similar to breaking down numbers into their prime factors. When factoring quadratic equations, or any polynomials, the goal is to decompose the expression into a product of simpler polynomials whose product gives back the original polynomial.

There are multiple factoring techniques, including:

• Greatest Common Factor (GCF) Factoring
• Factoring by Grouping
• Factoring Trinomials
• Difference of Squares

When presented with \( 5(x-9)(x-6)=0 \), we notice that the equation is already in a factored form, with \( x \) isolated within binomial factors. This reveals the roots directly through the Zero Product Property without additional factoring steps. Each binomial \( (x-9) \) and \( (x-6) \) represents potential solutions, which, when set to zero, allow us to solve for \( x \).

Algebraic Properties

Algebraic properties are rules that govern the operations and manipulation of algebraic expressions. These include the Commutative Property, Associative Property, Distributive Property, Identity Properties, and the Zero Product Property, which is particularly crucial for solving quadratic equations.

The Zero Product Property is an algebraic property stating that if \( ab=0 \), then either \( a=0 \) or \( b=0 \) or both. When applied to solving quadratic equations, it helps us decipher the roots by setting each factor of the equation to zero, as displayed in our original exercise.

By acknowledging and utilizing such properties, students can navigate through complex polynomial equations and arrive at solutions in a systematic and logical manner. Recognizing these properties not only aids in the computation but also in understanding the underlying principles of algebra that make the calculations valid.