To solve the equation:

\[
x^2 - 6x - 92 = -7x - 2
\]

### Step 1: Bring all terms to one side of the equation
Move the terms from the right side to the left side:
\[
x^2 - 6x - 92 + 7x + 2 = 0
\]
Combine like terms:
\[
x^2 + x - 90 = 0
\]

### Step 2: Solve the quadratic equation
The equation is now in the standard quadratic form:
\[
x^2 + x - 90 = 0
\]
We can solve this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 1 \), and \( c = -90 \).

### Step 3: Substitute the values into the formula
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-90)}}{2(1)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 360}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{361}}{2}
\]

### Step 4: Simplify the square root
\[
\sqrt{361} = 19
\]
So, we have:
\[
x = \frac{-1 \pm 19}{2}
\]

### Step 5: Solve for both possible values of \( x \)
1. \( x = \frac{-1 + 19}{2} = \frac{18}{2} = 9 \)
2. \( x = \frac{-1 - 19}{2} = \frac{-20}{2} = -10 \)

### Final Answer
The solutions to the equation are:
\[
x = 9 \quad \text{and} \quad x = -10
\]

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:
1. How can you verify the solutions \(x = 9\) and \(x = -10\) in the original equation?
2. What are the properties of quadratic equations when the discriminant is positive?
3. How would the solutions change if the constant term was different, like \(-92\) instead of \(-90\)?
4. Can you solve this quadratic using factoring instead of the quadratic formula?
5. What is the vertex of the parabola represented by \(x^2 + x - 90\)?

**Tip:** When solving a quadratic equation, the discriminant (\(b^2 - 4ac\)) helps determine the nature of the roots: positive for real and distinct, zero for real and equal, and negative for complex roots.

Learn how to solve the quadratic equation x^2 + x - 90 = 0 using the quadratic formula. This step-by-step guide explains the process, including combining like terms and simplifying the square root, making it perfect for students in Grades 9-10.

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