Problem 42 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 3 x^{2}-36 x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 12\).

Step 1: Identify the Type of Equation

The given equation is \(3x^2 - 36x = 0\). This is a quadratic equation because it has the form \( ax^2 + bx + c = 0\), where \(a = 3\), \(b = -36\), and \(c = 0\).

Step 2: Choose the Method

We can solve this quadratic equation either by factoring or using the Quadratic Formula. Since \(c = 0\), factoring can be the simplest approach here.

Step 3: Factoring the Quadratic Equation

Factor out the greatest common factor from the terms on the left side. Both terms have a factor of \(3x\), so we get: \[3x(x - 12) = 0\]

Step 4: Apply the Zero Product Property

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero: 1. \(3x = 0\)2. \(x - 12 = 0\)

Step 5: Solve Each Equation

For \(3x = 0\):Divide both sides by 3:\[x = 0\]For \(x - 12 = 0\):Add 12 to both sides:\[x = 12\]

Step 6: Verify the Solutions

Substitute \(x = 0\) and \(x = 12\) back into the original equation to verify:For \(x = 0\): \[3(0)^2 - 36(0) = 0\] which is true.For \(x = 12\): \[3(12)^2 - 36(12) = 0\] simplifies to \[432 - 432 = 0\] which is also true, confirming both solutions are correct.

Key Concepts

Factoring Quadratics

Factoring quadratics is often the go-to method when trying to solve quadratic equations, especially when the equation can be easily broken down into simpler elements. In our exercise, the equation provided is: \(3x^2 - 36x = 0\). Notice that both terms on the left side of the equation share a common factor, which is \(3x\).

Factoring involves:

• Identifying the Greatest Common Factor (GCF)
• Extracting the GCF from each term

Once we've identified \(3x\) as the GCF, we factor it out, arriving at: \[3x(x - 12) = 0\] This factored form makes it simple to apply the Zero Product Property in the subsequent steps, simplifying our path to the solution. Factoring is particularly handy when the quadratic can be expressed as a simple product of two binomials, making it a favored choice for this type of problem.

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation, even when factoring isn't straightforward. For any quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the formula is defined as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, the values of \(a\), \(b\), and \(c\) are the coefficients of the quadratic terms.

Using this formula involves:

• Substituting the coefficients into the formula
• Performing the arithmetic operations to find the values of \(x\)

While this method wasn't used in our particular solution (because factoring was simpler), it's extremely useful.

The Discriminant, found inside the square root (\(b^2 - 4ac\)), guides us as to the type of solutions we can expect:

• If it's positive, expect two real and distinct solutions.
• If zero, one real solution exists.
• If negative, no real solutions exist; instead, we have complex solutions.

Zero Product Property

The Zero Product Property is a fundamental concept when solving equations involving products. This property states that if the product of two factors is zero, then at least one of the factors must be zero. This rule is the key to unlocking solutions once we've factored the quadratic equation.

In our scenario, with the factored equation: \[3x(x - 12) = 0\] We apply the Zero Product Property by setting each factor equal to zero:

• \(3x = 0\) leads to \(x = 0\)
• \(x - 12 = 0\) leads to \(x = 12\)

Solving these simple linear equations gives us the solutions to the quadratic equation. This property is especially useful in breaking down complex problems into simpler ones, providing an intuitive path to the solution.