https://file.aiquickdraw.com/mathbot/file/1725889539535slxju8p9.jpg

The equation in the image is:

\[
\frac{x + 6}{x - 4} = \frac{1}{x + 1}
\]

Let's solve for \( x \) by cross-multiplying.

### Step 1: Cross-multiply the fractions

\[
(x + 6)(x + 1) = (x - 4)(1)
\]

### Step 2: Expand both sides

\[
(x + 6)(x + 1) = x^2 + x + 6x + 6 = x^2 + 7x + 6
\]
\[
(x - 4)(1) = x - 4
\]

Now the equation is:

\[
x^2 + 7x + 6 = x - 4
\]

### Step 3: Move all terms to one side to set the equation to zero

\[
x^2 + 7x + 6 - x + 4 = 0
\]

Simplifying:

\[
x^2 + 6x + 10 = 0
\]

### Step 4: Solve the quadratic equation

We will use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + 6x + 10 = 0\), the coefficients are:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 10 \)

Substitute into the quadratic formula:

\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(10)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 - 40}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{-4}}{2}
\]
\[
x = \frac{-6 \pm 2i}{2}
\]
\[
x = -3 \pm i
\]

### Final solution:
The solutions are \( x = -3 + i \) and \( x = -3 - i \), which are complex numbers.

Let me know if you'd like more details on any step!

Here are 5 related questions to expand on this topic:
1. How do we determine when to use the quadratic formula?
2. What does it mean when a quadratic equation has complex solutions?
3. How can we check if a quadratic equation has real or complex solutions without solving it?
4. How does the discriminant of a quadratic equation affect its solutions?
5. What are other methods for solving quadratic equations, and when are they useful?

**Tip:** The discriminant, \(b^2 - 4ac\), helps determine the nature of the solutions. If it's negative, the solutions are complex.

Solve for x: (x + 6) / (x - 4) = 1 / (x + 1)

Learn how to solve the equation (x + 6) / (x - 4) = 1 / (x + 1) step-by-step, using cross-multiplication and the quadratic formula. The solution involves complex numbers, making this problem suitable for students in Grades 10-12.

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