Problem 41 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 5 x^{2}-50 x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = 10\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). In this case, it is \(5x^2 - 50x = 0\).

Step 2: Factor out the Greatest Common Factor

Factor out the greatest common factor from the equation. Here, both terms have a common factor of 5x. \[ 5x(x - 10) = 0 \]

Step 3: Solve Each Factor

Set each factor equal to zero to find the solutions. 1. For the first factor: \(5x = 0\) - Divide by 5: \(x = 0\)2. For the second factor: \(x - 10 = 0\) - Add 10 to both sides: \(x = 10\)

Step 4: Verify the Solutions

Substitute \(x = 0\) and \(x = 10\) back into the original equation:- For \(x = 0\): \(5(0)^2 - 50(0) = 0\)- For \(x = 10\): \(5(10)^2 - 50(10) = 0\)Both check out, as they satisfy the original equation.

Key Concepts

Factoring

Factoring is a powerful method used to solve quadratic equations. When we factor, we are essentially reversing the process of expanding expressions. In this method, the goal is to express the quadratic equation in a product form. For example, given the equation \(5x^2 - 50x = 0\), we use factoring to simplify it.

In the factoring process, our first task is to look for common factors in all terms. It's like finding a common characteristic in numbers or expressions. In our example, both terms, \(5x^2\) and \(-50x\), share a common factor of \(5x\). Extracting this common factor transforms the original equation into:

• \(5x(x - 10) = 0\)

With the equation now factored, we can apply the zero-product property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. That's why we set each factor to zero to solve the equation:

• \(5x = 0\)
• \(x - 10 = 0\)

From these, we can solve for \(x\):

• If \(5x = 0\), then \(x = 0\).
• If \(x - 10 = 0\), then \(x = 10\).

Hence, the solutions to the equation are \(x = 0\) and \(x = 10\). Factoring is often the preferred method when it is easy to identify a common factor or simple binomials in the expression.

Quadratic Formula

The quadratic formula is a universal tool to solve any quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula derives from the process of completing the square and provides solutions for \(x\), even when the equation cannot be factored easily.

To apply the quadratic formula, identify the coefficients \(a\), \(b\), and \(c\) in your equation. In our equation, \(5x^2 - 50x = 0\),

• \(a = 5\)
• \(b = -50\)
• \(c = 0\)

Substituting these values into the quadratic formula, we have:\[x = \frac{-(-50) \pm \sqrt{(-50)^2 - 4 \cdot 5 \cdot 0}}{2 \cdot 5}\]Simplifying further will yield the solutions \(x = 0\) and \(x = 10\). The quadratic formula is especially useful when factoring is complex or not obvious, as it will always provide a solution assuming the discriminant (the part under the square root, \(b^2 - 4ac\)) is non-negative.

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is central in simplifying expressions and is a preliminary step in the factoring process. The GCF of a set of terms is the largest expression that divides each of the terms without leaving a remainder.

To find the GCF, factor each term into its prime components or look for the highest coefficient and variables common between them. In our quadratic equation example \(5x^2 - 50x = 0\), the terms are \(5x^2\) and \(-50x\). We can start by observing:

• Both terms have the variable \(x\). The smallest power of \(x\) that they both contain is \(x^1\).
• The coefficients are 5 and -50, and the largest number that divides both is 5.

Thus, the GCF here is \(5x\). By factoring out \(5x\) from the original expression, the equation simplifies, making it easier to solve:

• \(5x(x - 10) = 0\)

Identifying and factoring the GCF is crucial as it reduces the complexity, aligning the equation for simpler methods, like solving using the zero-product property.