Problem 103 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 2 x^{2}-7 x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(2x^2 - 7x = 0\) are \(x = 0\) and \(x = 7/2\).

Step 1: Set up the equation

The given equation is \(2x^2 - 7x = 0\). It's important to recognize that this is a type of quadratic equation, which could have two potential solutions.

Step 2: Factor out the GCD

The Greatest Common Divisor (GCD) of the two terms on the left-hand side of the equation is \(x\). Factoring out \(x\) gives: \(x(2x - 7) = 0\).

Step 3: Solve for x

Now you have a product of two terms equaling zero. The only way for a product to be zero is if at least one of the factors is zero. Thus, you set each factor equal to zero and solve for \(x\): \n\nCase 1: \(x = 0\) \n\nCase 2: \(2x - 7 = 0\), which simplifies to \(x = 7/2\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are a type of polynomial equation that can be represented in the general form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. These equations are characterized by the square of the variable \(x\) which results in a curve referred to as a parabola when graphed.
Solving quadratic equations involves finding the values of \(x\) that make the equation true. This can be accomplished through various methods:

• Factoring
• Completing the square
• Quadratic formula

The choice of method depends on the specific equation and its ease of factorization. In our example, the equation is solved conveniently using factoring due to the presence of a common factor and the equation's simple structure.

Factoring

Factoring is a way to express an equation as a product of its factors. This method is very useful in solving quadratic equations, especially when dealing with simple forms.
In the context of our equation \(2x^2 - 7x = 0\), factoring involves identifying a common factor that can be taken out from the terms involved.
Here, the greatest common factor between \(2x^2\) and \(-7x\) is \(x\). By factoring out \(x\), the equation becomes: \[x(2x - 7) = 0\] This step splits the quadratic equation into simpler expressions that can be individually evaluated.

Zero Product Property

The zero product property is a fundamental principle used in solving factored quadratic equations. This property states that if a product of two or more factors is zero, then at least one of the factors must be zero.
In mathematical terms, if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both).
Applied to our factored equation \(x(2x - 7) = 0\), we observe:

• \(x = 0\)
• \(2x - 7 = 0\)

Solving these, we find the solutions:

• \(x = 0\)
• \(x = \frac{7}{2}\)

Thus, the zero product property simplifies the process of finding solutions to an equation that has been factored.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest factor that divides two or more numbers. It plays a key role in simplifying algebraic expressions during the factoring process.
In solving quadratic equations like \(2x^2 - 7x = 0\), identifying the GCD helps to simplify the equation and make it more manageable.
Finding the GCD involves looking for the highest common factor that is shared by each term in the expression. For the equation provided, \(x\) is the GCD of \(2x^2\) and \(-7x\), and factoring it out leads to the expression \(x(2x - 7) = 0\).

• It simplifies the equation, making it easier to solve using the zero product property.
• It reduces computational complexity and errors.

Understanding the role of the GCD is essential for efficiently solving and simplifying equations in algebra.