Problem 22 - Applied Calculus

Question

Solve each equation by factoring. [Hint for: First factor out a fractional power.] $$ 3 x^{7 / 2}-12 x^{5 / 2}=36 x^{3 / 2} $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\), \(x = 6\), and \(x = -2\).

Step 1: Identify the Common Factor

We start by identifying the greatest common factor (GCF) in terms of fractional power of \(x\) across all terms. Each term in the equation contains a power of \(x\) that is a multiple of \(x^{3/2}\). Thus, the GCF is \(x^{3/2}\).

Step 2: Factor Out the Common Factor

Divide each term by \(x^{3/2}\) to factor it out:\[ 3x^{7/2} - 12x^{5/2} - 36x^{3/2} = x^{3/2}(3x^2 - 12x - 36) \]

Step 3: Set the Factored Equation to Zero

Now, set the equation equal to zero:\[ x^{3/2}(3x^2 - 12x - 36) = 0 \] We realize that solving this involves setting each factor to zero: \(x^{3/2} = 0\) and \(3x^2 - 12x - 36 = 0\).

Step 4: Solve for \(x\) when \(x^{3/2} = 0\)

Solving \(x^{3/2} = 0\) gives us the straightforward solution \(x = 0\).

Step 5: Solve the Quadratic Equation

Next, solve the quadratic equation \(3x^2 - 12x - 36 = 0\). Simplify by dividing the whole equation by 3:\[ x^2 - 4x - 12 = 0 \] This can be factored further into:\[ (x - 6)(x + 2) = 0 \]

Step 6: Solve for \(x\) from Each Factor

Setting each factor in the quadratic to zero gives:\(x - 6 = 0\) leading to \(x = 6\).\(x + 2 = 0\) leading to \(x = -2\).

Step 7: Combine Solutions

The solutions from the factored components are \(x = 0\), \(x = 6\), and \(x = -2\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in simplifying algebraic expressions and solving equations by factoring. It is the largest factor that divides two or more numbers or terms without leaving a remainder. In algebra, particularly when dealing with polynomial expressions, you often find this GCF to simplify equations.

Here's how you can identify the GCF in an equation with terms involving powers of a variable, like in the equation from the exercise:

• Look at the powers of the variables in each term.
• Identify the smallest power among these terms, as it can divide each of them evenly.
• Factor out the variable raised to this smallest power from each term in the expression.

In the example provided, each term had a power of the variable involving fractional exponents, such as \(x^{7/2}, x^{5/2}\), and \(x^{3/2}\). The GCF here is \(x^{3/2}\), as it is the smallest power that each term includes. This fundamental skill helps simplify equations and set the stage for further factorization and solving.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form \(ax^2 + bx + c = 0\). In the context of factoring, solving a quadratic equation involves expressing it as product of binomials, setting each to zero, and solving for the roots of the equation.

To factor and solve a quadratic equation:

• First, bring the equation to the standard form, if needed.
• Check for any common factors or simplify the equation if possible.
• Apply the quadratic formula, or if the equation can be factored easily, express it as two binomials: \((px + q)(rx + s) = 0\).
• Solve each binomial expression for \(x\).

In our exercise, after factoring out the GCF, the quadratic equation \(3x^2 - 12x - 36 = 0\) was reduced to \(x^2 - 4x - 12 = 0\). Factoring it into \((x - 6)(x + 2) = 0\), allowed us to solve for \(x\), yielding the solutions \(x = 6\) and \(x = -2\). Quadratic factoring is crucial because it transforms a two-step multiplication problem into a simple equation-solving task.

Fractional Exponents

Understanding fractional exponents is essential when working with algebraic expressions involving roots and powers. A fractional exponent like \(x^{m/n}\) is another way to express roots, where \(n\) is the root and \(m\) is the power.

To identify and manipulate fractional exponents:

• Recognize that \(x^{1/n}\) represents the \(n\)-th root of \(x\), such as \(x^{1/2}\) which is equivalent to \(\sqrt{x}\).
• A fractional exponent can simplify expressions just as whole number exponents do.
• Use the property \(x^{m/n} = (x^m)^{1/n}\) to either simplify or expand depending on the context.

In the given exercise, fractional exponents were present in all terms, guiding the factoring process. Recognizing \(x^{3/2}\) as the common factor was a vital step because it allowed us to simplify, resulting in a quadratic equation that was much easier to handle. Grasping the concept of fractional exponents will undoubtedly enhance your ability to manage complex algebraic expressions effectively.