Problem 93 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(2 x^{2}-7 x=0\)

Step-by-Step Solution

Answer

Therefore, the solutions for the given equation are \(x = 0\) and \(x = 7/2\).

Step 1: Factorisation

This equation can be factorised by taking out the common factor of x from each term. Hence, the factorized form of the given equation is \(x(2x - 7) = 0\).

Step 2: Apply the Zero Product Property

We now apply the Zero Product Property, setting each factor equal to zero. This results in the two equations: \(x = 0\) and \(2x - 7 = 0\).

Step 3: Solve the Equations

The first equation, \(x = 0\), is already solved. For the second equation \(2x - 7 = 0\), we add 7 to both sides to get \(2x = 7\), then divide both sides by 2 to solve for x, giving us \(x = 7/2\).

Key Concepts

Factorization

Factorization involves breaking down an algebraic expression into simpler factors. Think of it like your factor tree in mathematics but applied to algebraic expressions. When you have a quadratic equation like \(2x^2 - 7x = 0\), look for a common factor in the terms. In this case, \(x\) is common.

• Take \(x\) out from each term to get \(x(2x - 7) = 0\).

This process of factorizing equations simplifies them and makes solving them much easier. Simpler expressions are easier to manage and solve.

Zero Product Property

The Zero Product Property is a valuable concept when solving equations, especially factored forms. It states that if the product of two numbers is zero, at least one of the numbers must be zero.
Consider our factored equation \(x(2x - 7) = 0\). Applying the property means:

• \(x = 0\)
• \(2x - 7 = 0\)

This property essentially splits a single equation into two smaller ones. Each of these must be solved individually. It simplifies the approach to finding the roots of the equation by narrowing down the possibilities.

Solving Equations

Solving quadratic equations isn't as daunting when broken down into manageable steps. After factorization and applying the Zero Product Property, you will get simpler equations to solve.

• The first, \(x = 0\), is straightforward.
• For \(2x - 7 = 0\), you solve by isolating \(x\). Add 7 to both sides to obtain \(2x = 7\). Then, divide by 2 to solve for \(x\), resulting in \(x = \frac{7}{2}\).

Each step systematically narrows down possible solutions. This is a typical way to handle equations and it helps build your understanding of fundamental algebra.