Solve the equation by making an appropriate substitution.

left parenthesis x minus StartFraction 24 Over x EndFraction right parenthesis squared minus 3 left parenthesis x minus StartFraction 24 Over x EndFraction right parenthesis minus 10x−24x2−3x−24x−10equals=0

To solve the equation, we start by analyzing it and simplifying its structure. The equation appears to be a bit complex, but it can be broken down into manageable steps through substitution.

### Original equation:
\[
\left( x - \frac{24}{x} \right)^2 - 3 \left( x - \frac{24}{x} \right) - 10x - 24x^2 - 3x - 24x - 10 = 0
\]

#### Step 1: Simplify the terms outside of the substitution.
First, let's group the non-substitution terms for clarity:
\[
- 10x - 24x^2 - 3x - 24x - 10 = 0
\]
Simplify this part:
\[
-10x - 3x - 24x = -37x
\]
Thus, the equation becomes:
\[
\left( x - \frac{24}{x} \right)^2 - 3 \left( x - \frac{24}{x} \right) - 24x^2 - 37x - 10 = 0
\]

#### Step 2: Substitution.
Let \( y = x - \frac{24}{x} \). The equation simplifies to:
\[
y^2 - 3y - 24x^2 - 37x - 10 = 0
\]

Now, we need to handle both the terms with \(x\) and the expression in \(y\).

Since this equation mixes terms in \(x\) and \(y\), it becomes clear that further clarification is needed on the structure of the terms. Could you confirm if the expression is correct, or would you like assistance in adjusting the problem setup for the next steps?

Solve the equation by making an appropriate substitution.
(x - 24/x)^2 - 3(x - 24/x) - 10x - 24x^2 - 3x - 24x - 10 = 0

This guide walks through solving the complex equation (x - 24/x)^2 - 3(x - 24/x) - 10x - 24x^2 - 3x - 24x - 10 = 0 by using substitution and algebraic simplification. Suitable for students in Grades 9-12, it highlights key concepts like the substitution method and quadratic equations.

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