Problem 19 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$40 x-16 x^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{5}{2}\).

Step 1: Write the equation in standard form

The equation given is \(40x - 16x^2 = 0\). Re-order it in standard form, which is \(ax^2 + bx + c = 0\). In this case, it can be re-written as \(-16x^2 + 40x = 0\). For simplicity, you might further write it in positive leading coefficient form as \(16x^2 - 40x = 0\).

Step 2: Factor out the greatest common factor

First, identify the greatest common factor (GCF) of the terms in the equation. The GCF of \(16x^2\) and \(40x\) is \(8x\). Factor \(8x\) out of the equation: \(8x(2x - 5) = 0\).

Step 3: Use the Zero Product Property

Since the factored form is \(8x(2x - 5) = 0\), apply the zero product property which states that if a product of factors is zero, then at least one of the factors must be zero. Thus, set each factor equal to zero: \(8x = 0\) or \(2x - 5 = 0\).

Step 4: Solve each equation separately

Solve \(8x = 0\): Divide both sides by 8, giving \(x = 0\). Solve \(2x - 5 = 0\): Add 5 to both sides to get \(2x = 5\), then divide by 2 to find \(x = \frac{5}{2}\).

Step 5: State the solution

The solutions to the equation \(16x^2 - 40x = 0\) are \(x = 0\) and \(x = \frac{5}{2}\).

Key Concepts

Factoring Method

When it comes to solving quadratic equations, one of the most effective techniques you can use is the factoring method. A quadratic equation is typically expressed in the standard form as \( ax^2 + bx + c = 0 \). Factoring is suitable when the quadratic expression can be broken down into simpler multiplicative terms. These terms are known as factors, and rearranging the equation in such a manner allows us to find the values of \( x \) where the equation is true.

To factor a quadratic equation:

• First, ensure the equation is in standard form.
• The next step is finding factors of the quadratic expression. This means rewriting the quadratic as a product of two binomials.
• Finally, set each binomial equal to zero and solve for \( x \). This is possible thanks to the Zero Product Property, which we'll dive into later in this article.

Keep in mind that factoring may not always be straightforward, particularly if the quadratic doesn't exhibit easily recognizable patterns. Nevertheless, practicing this method with various equations will improve your proficiency over time.

Greatest Common Factor

When factoring quadratic equations, identifying the greatest common factor (GCF) can greatly simplify the process. The GCF is the largest number or expression that divides the terms of the quadratic equation without a remainder.

Here’s how to find and use the GCF:

• Examine the coefficients and variables in the terms of the quadratic equation.
• Determine the highest number or expression common to all the terms. This becomes your GCF.
• Factor the GCF out of the entire equation. This simplifies the equation and makes further factoring easier.

For instance, given \(16x^2 - 40x = 0\), the GCF is \(8x\). By factoring \(8x\) out, we simplify it to \(8x(2x - 5) = 0\). Recognizing and factoring out the GCF early on leads to a much simpler equation to work with and sets the stage for applying the zero product property.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that helps us solve equations, especially when using the factoring method. According to this property, if the product of two or more factors is zero, then at least one of the factors must be zero. This is an incredibly useful characteristic, as it allows us to solve polynomial equations once they are factored.

Using the zero product property involves a few key steps:

• First, ensure the quadratic equation is factored completely.
• Set each factor equal to zero separately.
• Solve each resulting equation to find the values of \( x \).

For example, after factoring the equation \(16x^2 - 40x = 0\) into \(8x(2x - 5) = 0\), you would use the Zero Product Property. This means setting both \(8x = 0\) and \(2x - 5 = 0\) to find \(x = 0\) and \(x = \frac{5}{2}\). This reveals the solutions to the original quadratic equation.

Understanding this property helps demystify why factoring is such a potent technique for solving quadratic equations.