<Question>
Compute $a+b+c,$ given that $a,$ $b,$ and $c$ are the roots of \[\frac{1}{x} + 5x^2 = 6x - 24.\]
</Question>

<think>
<Plan_1>
Okay, let's see. I need to compute a + b + c, where a, b, and c are the roots of the equation given: 1/x + 5x² = 6x - 24. Hmm, first, I remember that for polynomial equations, there's something called Vieta's formulas which relates the coefficients of the polynomial to the sums and products of its roots. But the equation here isn't in the standard polynomial form. So, maybe I need to rearrange the given equation into a standard polynomial equation first?

Let me write down the equation again to make sure I have it right: 1/x + 5x² = 6x - 24. So, the left side is 1/x plus 5x squared, and the right side is 6x minus 24.

First step, perhaps multiply both sides by x to eliminate the fraction. Let me try that.
Multiplying every term by x gives: 1 + 5x³ = 6x² - 24x.

Now, let's move all terms to one side to set the equation to zero.
Subtract 6x² and add 24x to both sides: 1 + 5x³ - 6x² + 24x = 0.

Let me rearrange the terms in descending order of degree: 5x³ - 6x² + 24x + 1 = 0.

So, the equation is a cubic: 5x³ - 6x² + 24x + 1 = 0.

Now, for a cubic equation of the form ax³ + bx² + cx + d = 0, Vieta's formula tells us that the sum of the roots is -b/a. In this case, the coefficient of x³ is 5, and the coefficient of x² is -6. So, the sum of the roots a + b + c should be -(-6)/5 = 6/5.

Wait, is that right?

But hold on, let me make sure I didn't make any mistakes in the algebraic manipulation earlier.

<prompt_1.1> Verify the rearrangement of terms in the polynomial equation. </prompt_1.1>

So, according to Vieta's formula, the sum of the roots is indeed 6/5.

But wait, the original equation had a 1/x term. Does that affect anything?

<prompt_1.2> Consider the impact of the original equation's terms on the polynomial conversion and Vieta's applicability. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Starting from the original equation: 1/x + 5x² = 6x - 24. Multiplying both sides by x: 1 + 5x³ = 6x² - 24x. Then, subtract 6x² and add 24x: 5x³ - 6x² + 24x + 1 = 0.
</execution_1.1>

<execution_1.2>
Because when I multiplied both sides by x, I transformed the equation into a cubic, which is a polynomial equation. Since the original equation had a term with 1/x, does that mean that x=0 is a possible root? But in the original equation, substituting x=0 would make 1/x undefined, so x=0 is not in the domain of the original equation. Therefore, the roots a, b, c must all be non-zero. So, when I multiplied both sides by x, I didn't introduce any extraneous roots because x=0 isn't a valid solution here.

Therefore, the cubic equation 5x³ - 6x² + 24x + 1 = 0 has three roots a, b, c, none of which are zero. Then Vieta's formula applies here. So, the sum of the roots is 6/5.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Wait, but let me recall Vieta's formula for cubic equations.

<prompt_2.1> Reaffirm the application of Vieta's formula for the cubic equation. </prompt_2.1>

Therefore, the answer should be 6/5.

But let me check again.

<prompt_2.2> Confirm the solution through multiple checks of the algebraic manipulation and Vieta's formula. </prompt_2.2>

Is there a possibility that the original equation could have been interpreted differently? For example, maybe the equation is 1/(x + 5x²) = 6x - 24? But that's not how it's written.

<prompt_2.3> Consider alternative interpretations of the original equation and confirm the correct approach. </prompt_2.3>

Alternatively, suppose I wanted to check if there's a mistake in the problem setup. For example, maybe the equation was supposed to be 1/(x + 5x²) = 6x -24, but that's not the case here.

Alternatively, maybe there's a typo in the original problem, but assuming it's correct as given, then the steps are right.

Alternatively, maybe I can test with sample roots, but since we don't have the actual roots, that's not possible.

Alternatively, perhaps I can use another method to confirm.

<prompt_2.4> Explore alternative methods to confirm the solution and conclude the problem. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Given a cubic equation ax³ + bx² + cx + d = 0, the sum of the roots is -b/a. Here, a is 5, b is -6. So, sum of roots is -(-6)/5 = 6/5. That seems correct.

But let me make sure that the cubic equation is indeed 5x³ -6x² +24x +1 = 0. Yes, from the original equation:

1/x + 5x² = 6x -24.

Multiply both sides by x: 1 + 5x³ = 6x² -24x.

Bring all terms to left: 5x³ -6x² +24x +1 = 0.
</execution_2.1>

<execution_2.2>
The standard form is ax³ + bx² + cx + d = 0. Then, the sum of roots is -b/a. Here, a is 5, b is -6. So, sum is -(-6)/5 = 6/5.
</execution_2.2>

<execution_2.3>
The original equation is written as 1/x + 5x² = 6x - 24, so it's 1 divided by x plus 5x squared equals 6x minus 24. So, there's no ambiguity in the left-hand side. So, multiplying by x is the correct approach.

Alternatively, if someone misapplies the multiplication by x, but in this case, we did it correctly.
</execution_2.3>

<execution_2.4>
Let me think. Suppose the equation is 5x³ -6x² +24x +1 =0. If I use Vieta's formula, then the sum is 6/5. So, unless there's a miscalculation in the coefficients, that's the answer.

Wait, let's rederive the equation again step by step.

Original equation: 1/x +5x² =6x -24.

Multiply both sides by x: 1 +5x³ =6x² -24x.

Bring all terms to left side: 5x³ -6x² +24x +1 =0.

Yes, that's correct. So, the cubic equation is correct. Then, Vieta's formula: sum of roots is -b/a. Here, a is coefficient of x³ which is 5, b is coefficient of x² which is -6. So, sum is -(-6)/5 =6/5.
</execution_2.4>
</Execution_2>

<Final_answer>
Therefore, the sum of the roots $a + b + c$ is $\boxed{\frac{6}{5}}$.
</Final_answer>
</think>