Problem 19 - Applied Calculus

Question

Solve each equation by factoring. [Hint for: First factor out a fractional power.] $$ 3 x^{5 / 2}-6 x^{3 / 2}=9 x^{1 / 2} $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Rewrite the Equation

The given equation is \(3x^{5/2} - 6x^{3/2} = 9x^{1/2}\). Notice that all terms have a common factor related to powers of \(x\). In this case, \(x^{1/2}\) is the common term.

Step 2: Factor Out the Greatest Common Factor

Factor out \(x^{1/2}\) from each term:\[ x^{1/2}(3x^2 - 6x) = 9x^{1/2} \]

Step 3: Simplify Both Sides

Divide both sides of the equation by \(x^{1/2}\) to isolate the quadratic expression. This simplifies to:\[ 3x^2 - 6x = 9 \]

Step 4: Move All Terms to One Side

To solve the equation by factoring, move all terms to one side of the equation:\[ 3x^2 - 6x - 9 = 0 \]

Step 5: Factor the Quadratic

Factor the quadratic equation. First, factor out the greatest common divisor (GCD) for simplicity:\[ 3(x^2 - 2x - 3) = 0 \]Then, factor \(x^2 - 2x - 3\):\[ x^2 - 2x - 3 = (x - 3)(x + 1) \] The equation becomes:\[ 3(x - 3)(x + 1) = 0 \]

Step 6: Solve for the Roots

Set each factor equal to zero and solve:\[ x - 3 = 0 \Rightarrow x = 3 \]\[ x + 1 = 0 \Rightarrow x = -1 \]

Step 7: Check for Restrictions

Since we divided by \(x^{1/2}\), which is equivalent to \(x^{1/2} eq 0\), ensure that \(x > 0\). Therefore, only \(x = 3\) is valid.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( x \) represents an unknown variable. Quadratic equations are central in algebra and their solutions are called roots. Quadratic equations often appear in problems related to areas, projectile motions, and optimization scenarios.
Quadratic equations can be solved through various methods:

• Factoring: This involves expressing the quadratic in a product of its factors, which can then be set to zero to find the roots.
• Completing the square: This method involves rearranging the equation to form a perfect square trinomial.
• Quadratic formula: A general formula that provides solutions to any quadratic equation, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In our problem, we used factoring after rearranging and simplifying the terms.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest factor that two or more numbers have in common. It is a helpful tool in simplifying equations, particularly when you want to break down expressions to their simplest forms.
In the given exercise, the GCF was used to factor out common terms from the equation. First, observe that each term in the original equation \(3x^{5/2} - 6x^{3/2} = 9x^{1/2}\) contains a power of \(x\). Here, \(x^{1/2}\) is identified as the GCF. By factoring \(x^{1/2}\) from all terms, the equation is simplified, allowing easier manipulation and solving of the quadratic form.
Understanding how to find and factor the GCF is key to solving equations more efficiently, especially in algebraic expressions.

Roots of Equation

The roots of an equation are the values of \(x\) that satisfy the equation, making it true. For quadratic equations, there are typically two roots, but they can be real or complex. In some special cases, if the discriminant \((b^2 - 4ac)\) is zero, both roots can be the same.
The process to calculate these roots often depends on the method used to solve the quadratic. In our exercise, after factoring the equation, we found the roots by setting each factor equal to zero:

• \(x - 3 = 0\) gives \(x = 3\)
• \(x + 1 = 0\) gives \(x = -1\)

It’s important to consider any constraints from the problem. For this exercise, since we initially factored out a term with a fractional power, addition checks are necessary to confirm that results meet any necessary conditions, such as keeping \(x > 0\) due to division by a fractional power term.

Fractional Powers

Fractional powers refer to exponents that are fractions, and they represent both a root and a power. For example, \(x^{1/2}\) means the square root of \(x\), and \(x^{3/2}\) means \(x\) raised to the power of 3 and then taking the square root.
In our exercise, fractional powers simplify the expression by factoring out the smallest fractional power, which here was \(x^{1/2}\). This manipulation simplifies the equation and helps isolate terms, making it easier to solve. Using fractional powers effectively in algebra requires a solid understanding of both roots and exponents.
When solving equations involving fractional powers, it is crucial to remember that such operations imply certain constraints, such as keeping the values of \(x\) within domains where they are defined and valid.