https://file.aiquickdraw.com/mathbot/file/17260134319702l2pmztu.jpg

Let's break down the equations from the image into steps to transform them into quadratic equations.

### Equation 6:
\[
\frac{1}{x} - \frac{x}{6} = \frac{2}{3}
\]

To solve this, follow these steps:

1. Find a common denominator for the terms on the left-hand side. The common denominator between \( x \) and \( 6 \) is \( 6x \).

   \[
   \frac{1}{x} - \frac{x}{6} = \frac{6}{6x} - \frac{x^2}{6x}
   \]

   This gives:
   \[
   \frac{6 - x^2}{6x} = \frac{2}{3}
   \]

2. Cross-multiply to eliminate the fractions:
   \[
   3(6 - x^2) = 12x
   \]

   Simplifying:
   \[
   18 - 3x^2 = 12x
   \]

3. Rearrange this into standard quadratic form:
   \[
   3x^2 + 12x - 18 = 0
   \]

   Finally, divide through by 3 to simplify:
   \[
   x^2 + 4x - 6 = 0
   \]

### Equation 7:
\[
\frac{4}{t - 3} + \frac{t}{2} = -2
\]

To solve this, follow these steps:

1. Find a common denominator for the fractions on the left-hand side. The common denominator between \( t - 3 \) and \( 2 \) is \( 2(t - 3) \).

   Multiply both terms by \( 2(t - 3) \) to eliminate the fractions:
   \[
   2(t - 3)\left(\frac{4}{t - 3}\right) + 2(t - 3)\left(\frac{t}{2}\right) = 2(t - 3)(-2)
   \]

   This simplifies to:
   \[
   8 + t(t - 3) = -4(t - 3)
   \]

2. Expand both sides:
   \[
   8 + t^2 - 3t = -8 + 4t
   \]

3. Rearrange everything to one side:
   \[
   t^2 - 3t - 4t + 8 + 8 = 0
   \]

   Simplifying:
   \[
   t^2 - 7t + 16 = 0
   \]

Now, both quadratic equations are:

- Equation 6: \( x^2 + 4x - 6 = 0 \)
- Equation 7: \( t^2 - 7t + 16 = 0 \)

Would you like further details or explanations on any of these steps?

### Related Questions:
1. What are the solutions to the quadratic equation \( x^2 + 4x - 6 = 0 \)?
2. How do you solve quadratic equations using the quadratic formula?
3. What is the discriminant of the quadratic \( t^2 - 7t + 16 = 0 \), and what does it tell you about the nature of the roots?
4. How can you factor a quadratic equation if it's factorable?
5. What are some methods for solving quadratic equations besides factoring?

### Tip:
When solving equations with fractions, clearing the denominators early by multiplying both sides can simplify the process significantly.

Transform the following into quadratic equation: 1/x - x/6 = 2/3 and 4/(t - 3) + t/2 = -2.

This problem involves transforming fractional equations into quadratic equations. Learn step-by-step how to simplify and solve equations such as 1/x - x/6 = 2/3 and 4/(t - 3) + t/2 = -2, using cross-multiplication and quadratic principles. Suitable for Grades 9-10.

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