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transform first the equations below into quadratic equations in standard form.    please provide every solutions 

To transform the given equation into a quadratic equation in standard form, let's start by carefully analyzing and solving it step by step.

### Given equation:
\[
\frac{5 - x + 2}{4x} = \frac{x - 1}{3}
\]

### Step 1: Eliminate the fractions by cross-multiplying.

Multiply both sides by \(12x\) (which is the least common multiple of \(4x\) and \(3\)):

\[
12x \cdot \frac{5 - x + 2}{4x} = 12x \cdot \frac{x - 1}{3}
\]

This simplifies to:

\[
3(7 - x) = 4x(x - 1)
\]

### Step 2: Expand both sides.

Expand the left and right sides:

\[
21 - 3x = 4x^2 - 4x
\]

### Step 3: Move all terms to one side to form a quadratic equation.

Add \(3x\) and subtract \(21\) from both sides:

\[
0 = 4x^2 - 4x + 3x - 21
\]

Simplify further:

\[
4x^2 - x - 21 = 0
\]

### Final Quadratic Equation (in standard form):

\[
4x^2 - x - 21 = 0
\]

This is the quadratic equation in standard form derived from the given problem.

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Would you like more details on this process or have any other questions? Here are five related questions you might find interesting:

1. How do you solve this quadratic equation \(4x^2 - x - 21 = 0\) using the quadratic formula?
2. What are the real or complex roots of the quadratic equation \(4x^2 - x - 21 = 0\)?
3. Can this quadratic equation be factored easily?
4. How does the discriminant help determine the nature of the roots for the quadratic equation \(4x^2 - x - 21 = 0\)?
5. What are the possible applications of quadratic equations in real-world scenarios?

**Tip:** When transforming equations, always start by eliminating fractions or decimals to simplify the process of rearranging terms.

Learn how to transform equations into quadratic form with step-by-step instructions. This problem involves eliminating fractions, applying cross-multiplication, and forming quadratic equations in standard form. Suitable for high school students.

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