Solve this using EQUATION IN QUADRATIC FORM: 1.) x/3 + 12/x = x-2    

To solve the equation \( \frac{x}{3} + \frac{12}{x} = x - 2 \) using the quadratic form, follow these steps:

### Step 1: Eliminate the fractions by multiplying the entire equation by \( x \times 3 \).
Multiply each term by \( 3x \) to clear the fractions:

\[
3x \times \left(\frac{x}{3}\right) + 3x \times \left(\frac{12}{x}\right) = 3x \times (x - 2)
\]

Simplifying, we get:

\[
x^2 + 36 = 3x^2 - 6x
\]

### Step 2: Rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \).
Move all terms to one side of the equation:

\[
x^2 + 36 - 3x^2 + 6x = 0
\]

Combine like terms:

\[
-2x^2 + 6x + 36 = 0
\]

### Step 3: Multiply the entire equation by \(-1\) (to make the quadratic coefficient positive):
\[
2x^2 - 6x - 36 = 0
\]

### Step 4: Simplify the equation by dividing by the greatest common factor (GCF).
The GCF of the coefficients 2, -6, and -36 is 2. Divide the entire equation by 2:

\[
x^2 - 3x - 18 = 0
\]

### Step 5: Solve the quadratic equation using the quadratic formula.
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -3 \), and \( c = -18 \). Plugging in these values:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-18)}}{2(1)}
\]

\[
x = \frac{3 \pm \sqrt{9 + 72}}{2}
\]

\[
x = \frac{3 \pm \sqrt{81}}{2}
\]

\[
x = \frac{3 \pm 9}{2}
\]

### Step 6: Find the solutions.
This gives us two possible solutions:

1. \( x = \frac{3 + 9}{2} = \frac{12}{2} = 6 \)
2. \( x = \frac{3 - 9}{2} = \frac{-6}{2} = -3 \)

So, the solutions are \( x = 6 \) and \( x = -3 \).

### Final Answer:
The solutions to the equation \( \frac{x}{3} + \frac{12}{x} = x - 2 \) are \( x = 6 \) and \( x = -3 \).

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. How would you verify that \( x = 6 \) and \( x = -3 \) are indeed solutions to the original equation?
2. Can a quadratic equation have more than two real solutions?
3. How does the discriminant of a quadratic equation determine the nature of its roots?
4. How do you solve a quadratic equation by completing the square instead of using the quadratic formula?
5. What happens if the quadratic equation has complex roots? How would you interpret them?

**Tip:** Always check the solutions by substituting them back into the original equation to ensure they satisfy it.

Solve this using EQUATION IN QUADRATIC FORM: 1.) x/3 + 12/x = x-2

Discover how to solve the equation x/3 + 12/x = x-2 by transforming it into a quadratic equation. This step-by-step guide covers key algebraic concepts, including equation manipulation and the quadratic formula, making it ideal for students in Grades 9-11.

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