Problem 42 - 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 Equation Type

The given equation is a quadratic equation in standard form: \(3x^2 - 36x = 0\). We notice that it is a quadratic equation without a constant term.

Step 2: Factor Out the Greatest Common Factor

To simplify the equation, first factor out the greatest common factor (GCF) from all terms. The GCF of \(3x^2\) and \(36x\) is \(3x\). Thus, apply the factorization: \(3x(x - 12) = 0\).

Step 3: Apply the Zero Product Property

Once the equation is factored, apply the zero product property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Therefore, set each factor equal to zero: \(3x = 0\) and \(x - 12 = 0\).

Step 4: Solve for x

Solving each equation from the factored form: 1. \(3x = 0\) leads to \(x = 0\).2. \(x - 12 = 0\) leads to \(x = 12\).

Step 5: Verify the Solutions

Substitute the solutions back into the original equation to check. - For \(x = 0\): \(3(0)^2 - 36(0) = 0\) holds true.- For \(x = 12\): \(3(12)^2 - 36(12) = 432 - 432 = 0\), which is true.

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations by breaking down expressions into simpler factors. It is especially useful when the equations can be easily rewritten as a product of terms. In the example problem, the quadratic equation is given as \(3x^2 - 36x = 0\). The first step in factoring is to identify the greatest common factor (GCF) of each term in the equation.

Here’s how you factor this equation:

• Identify the GCF in the terms \(3x^2\) and \(-36x\). Here, the GCF is \(3x\).
• Factor out the GCF from the equation: \(3x(x - 12) = 0\).

This process transforms the equation, making it easier to solve. Factoring lets us split the equation into distinct terms, which individually can equal zero. Remember, factoring only works well when the equation has whole number factors.

Quadratic Formula

The quadratic formula is a powerful tool used for solving quadratic equations that are not easily factorable. It's most useful when the equation takes the form \(ax^2 + bx + c = 0\), but the factors are not obvious or don't yield integers. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use it, first identify the coefficients \(a\), \(b\), and \(c\) from your equation. Substitute them into the formula to calculate the roots or solutions.

In the case of our original problem, although the quadratic formula is not needed due to easy factoring, knowing it as an alternative method is vital:

• Helps when factoring is too complex or terms do not factor neatly.
• Provides a general solution applicable to any quadratic equation.
• Can confirm if roots are real or complex through the discriminant \(b^2 - 4ac\).

For more complex equations where factoring is impractical, the quadratic formula is your go-to solution strategy.

Zero Product Property

The zero product property is a fundamental concept used after factoring expressions. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This is crucial in solving factored equations.

For example, with our equation factored as \(3x(x - 12) = 0\):

• The zero product property allows you to set each factor separately to zero: \(3x = 0\) or \(x - 12 = 0\).
• Solve these equations for \(x\) to find the possible values.
• Once simplified, you find \(x = 0\) from \(3x = 0\), and \(x = 12\) from \(x - 12 = 0\).

By applying this property, we turn a single complex equation into simpler ones. This makes finding solutions straightforward, providing an efficient conclusion.