If $x - \frac 3 x = 6,\; x \ne 0,$ then the value of $\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}$ is:

Question

If $x - \frac 3 x = 6,\; x \ne 0,$  then the value of  $\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}$  is:

Prepp Team · Accepted Answer

Understanding the Algebraic Problem

The problem asks us to find the value of a specific algebraic expression given an initial equation. We are given the equation $x - \frac 3 x = 6$, with the condition that $x \ne 0$. We need to evaluate the expression $\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}$.

To solve this, we should try to manipulate the given equation and the expression to find relationships that simplify the calculation. Often, problems like this involve algebraic identities or clever substitutions.

Step-by-Step Solution

Let's start by analyzing the given equation:

$x - \frac 3 x = 6$

Since $x \ne 0$, we can multiply the entire equation by $x$ to eliminate the fraction:

$x(x - \frac 3 x) = 6x$

$x^2 - 3 = 6x$

This gives us a useful relationship: $x^2 - 3 = 6x$. We can also rearrange this as $x^2 - 6x - 3 = 0$.

Analyzing the Denominator

The denominator of the expression is $x^2 - 3x - 3$. Let's try to simplify this using the relationship $x^2 - 3 = 6x$:

$x^2 - 3x - 3 = (x^2 - 3) - 3x$

Substitute $x^2 - 3 = 6x$ into this:

$(x^2 - 3) - 3x = 6x - 3x = 3x$

So, the denominator simplifies to $3x$.

Analyzing the Numerator

The numerator of the expression is $x^4 - \frac {27}{x^2}$. Let's try to relate this to the original equation or derived relationships.

Consider cubing the original equation $x - \frac 3 x = 6$. Recall the identity $(a-b)^3 = a^3 - b^3 - 3ab(a-b)$. Here, $a=x$ and $b=\frac 3 x$.

$(x - \frac 3 x)^3 = 6^3$

$x^3 - (\frac 3 x)^3 - 3(x)(\frac 3 x)(x - \frac 3 x) = 216$

$x^3 - \frac {27}{x^3} - 9(x - \frac 3 x) = 216$

Substitute the original equation $x - \frac 3 x = 6$ into this:

$x^3 - \frac {27}{x^3} - 9(6) = 216$

$x^3 - \frac {27}{x^3} - 54 = 216$

$x^3 - \frac {27}{x^3} = 216 + 54$

$x^3 - \frac {27}{x^3} = 270$

Now let's look at the numerator again: $x^4 - \frac {27}{x^2}$. Can we express this using $x^3 - \frac {27}{x^3}$?

Let's factor $x$ from the numerator term:

$x^4 - \frac {27}{x^2} = x \left(x^3 - \frac {27}{x^3}\right)$

We just found that $x^3 - \frac {27}{x^3} = 270$. Substitute this value:

$x \left(x^3 - \frac {27}{x^3}\right) = x (270) = 270x$

So, the numerator simplifies to $270x$.

Evaluating the Expression

Now we have the simplified numerator and denominator:

Numerator: $270x$
Denominator: $3x$

The expression is $\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3} = \frac{270x}{3x}$.

Since the problem states $x \ne 0$, we can cancel $x$ from the numerator and the denominator:

$\frac{270x}{3x} = \frac{270}{3}$

$\frac{270}{3} = 90$

Thus, the value of the expression is 90.

Summary of Steps

Start with the given equation $x - \frac 3 x = 6$.
Manipulate the equation to find useful relationships, like $x^2 - 3 = 6x$.
Simplify the denominator $x^2 - 3x - 3$ using the derived relationship.
Cube the original equation to find the value of $x^3 - \frac{27}{x^3}$.
Factor the numerator $x^4 - \frac {27}{x^2}$ and use the value of $x^3 - \frac{27}{x^3}$.
Substitute the simplified numerator and denominator into the expression and evaluate.

Checking the Options

The calculated value is 90, which matches one of the given options.

Revision Table: Key Algebraic Manipulations

Starting Point	Manipulation	Result
$x - \frac 3 x = 6$	Multiply by $x$	$x^2 - 3 = 6x$
$x - \frac 3 x = 6$	Cube both sides	$x^3 - \frac {27}{x^3} - 9(x - \frac 3 x) = 216$
$x^3 - \frac {27}{x^3} - 9(x - \frac 3 x) = 216$	Substitute $x - \frac 3 x = 6$	$x^3 - \frac {27}{x^3} = 270$
Denominator: $x^2 - 3x - 3$	Use $x^2 - 3 = 6x$	$6x - 3x = 3x$
Numerator: $x^4 - \frac {27}{x^2}$	Factor $x$ and use $x^3 - \frac {27}{x^3} = 270$	$x(x^3 - \frac {27}{x^3}) = x(270) = 270x$

Additional Information on Algebraic Expressions and Equations

This problem demonstrates the power of algebraic manipulation. By creatively rearranging and transforming the given equation, we were able to simplify a complex expression without needing to find the specific value of $x$.

Equation: A statement that two mathematical expressions are equal. Solving an equation means finding the value(s) of the variable(s) that make the statement true.
Expression: A combination of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, division). An expression does not contain an equals sign.
Algebraic Manipulation: The process of rearranging or transforming algebraic expressions or equations while maintaining their equality or equivalence. This includes operations like adding or subtracting the same value from both sides, multiplying or dividing both sides by the same non-zero value, factoring, expanding, etc.
Identities: Equations that are true for all values of the variables for which both sides of the equation are defined. The identity $(a-b)^3 = a^3 - b^3 - 3ab(a-b)$ was crucial in this problem.
Substitution: Replacing a variable or an expression with another expression that it is equal to. This was used multiple times in the solution process.

Problems like this reinforce the importance of recognizing algebraic patterns and knowing how to apply standard identities to simplify complex expressions and solve equations efficiently.

Prepp Team · Answer

54

Prepp Team · Answer

270

Prepp Team · Answer

80

Starting Point	Manipulation	Result
\(x - \frac 3 x = 6\)	Multiply by \(x\)	\(x^2 - 3 = 6x\)
\(x - \frac 3 x = 6\)	Cube both sides	\(x^3 - \frac {27}{x^3} - 9(x - \frac 3 x) = 216\)
\(x^3 - \frac {27}{x^3} - 9(x - \frac 3 x) = 216\)	Substitute \(x - \frac 3 x = 6\)	\(x^3 - \frac {27}{x^3} = 270\)
Denominator: \(x^2 - 3x - 3\)	Use \(x^2 - 3 = 6x\)	\(6x - 3x = 3x\)
Numerator: \(x^4 - \frac {27}{x^2}\)	Factor \(x\) and use \(x^3 - \frac {27}{x^3} = 270\)	\(x(x^3 - \frac {27}{x^3}) = x(270) = 270x\)