Problem 11 - Algebra and Trigonometry

Question

Solve equation by factoring. $$ 2 x(x-3)=5 x^{2}-7 x $$

Step-by-Step Solution

Answer

The solutions to the equation are x = 0 and x = 1/3.

Step 1: Simplify the Equation

Simplify the equation by expanding the terms and bringing all the terms to one side. So, \(2x^2-6x=5x^2-7x\) simplifies to \(3x^2-x=0\)

Step 2: Factoring the Equation

Factor out an x from each term on the left side of the equation. This would look like \(x(3x-1) = 0\)

Step 3: Solve for x

Set both factors equal to zero and solve for x. So, \(x = 0\) or \(3x-1 = 0\) gives us the solutions \(x = 0\) or \(x = 1/3\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a fundamental technique in algebra to solve for the roots of the equation, which are the values of the variable that make the equation true.

• Standard Form: A quadratic equation is typically written in the standard form \( ax^2 + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are constants, and \( x \) is the variable.
• Factoring Process: To factor a quadratic equation, we look to express it as a product of two binomials, like \( (dx+e)(fx+g) = 0 \) . The constant terms \( e \) and \( g \) when multiplied should give \( ac \) , and when added, they should give \( b \) .
• Implementing in Exercise: In the given problem, the quadratic \( 3x^2 - x \) was factored to \( x(3x - 1) = 0 \) . This step breaks down the original quadratic into factors that can easily be solved, revealing the roots of the equation.

Factoring is critical for simplifying complex quadratic equations into more manageable pieces that satisfy the zero product property, which we utilize to find the equation’s solutions.

Zero Product Property

The zero product property is a key principle in algebra that states if the product of two expressions is zero, then at least one of the expressions must be zero.

• Algebraic Expression: For instance, if we have \( ab = 0 \) , then \( a = 0 \) or \( b = 0 \) , or both \( a \) and \( b \) could be zero.
• Application: When a quadratic equation is factored, we set each factor equal to zero. This application of the zero product property allows us to solve for the variable.
• Zero Product in Exercise: In the exercise, the factored form was \( x(3x - 1) = 0 \) . By the zero product property, we set each factor \( x = 0 \) and \( 3x - 1 = 0 \) to find the solutions of the original equation.

The zero product property is a simple yet profound rule that provides a straightforward path to finding the solutions to quadratic equations once they’ve been factored.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing an expression to its simplest form by performing all possible arithmetic operations and combining like terms.

• Combining Like Terms: In algebra, like terms are terms that contain the same variables raised to the same power. For example, \( 2x \) and \( 5x \) are like terms whereas \( 2x \) and \( 5x^2 \) are not.
• Incorporating Simplification: Before factoring, it can be necessary to simplify an equation by combining like terms or moving all terms to one side of the equation, which often involves distributing and combining similar terms.
• Exercise Context: In the example, the equation \( 2x(x-3) = 5x^2 - 7x \) was simplified to \( 3x^2 - x = 0 \) through expansion and combination of like terms, setting the stage for the factoring process.

Simplifying expressions is a foundational skill for solving equations, as it makes patterns more evident, thus often unveiling the pathway to the solution.