To solve the equation \(-18 + 6x - 7x = -5x + 2x^2\), we need to simplify and then solve for \( x \).

Here are the steps:

1. Combine like terms on both sides of the equation:
   \[
   -18 + 6x - 7x = -5x + 2x^2
   \]
   \[
   -18 - x = -5x + 2x^2
   \]

2. Move all terms to one side to set the equation to zero:
   \[
   0 = 2x^2 - 5x + x - 18
   \]
   \[
   2x^2 - 4x - 18 = 0
   \]

3. Simplify the equation by dividing all terms by 2:
   \[
   x^2 - 2x - 9 = 0
   \]

4. Solve the quadratic equation \( x^2 - 2x - 9 = 0 \) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = -2 \), and \( c = -9 \).

   \[
   x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-9)}}{2(1)}
   \]
   \[
   x = \frac{2 \pm \sqrt{4 + 36}}{2}
   \]
   \[
   x = \frac{2 \pm \sqrt{40}}{2}
   \]
   \[
   x = \frac{2 \pm 2\sqrt{10}}{2}
   \]
   \[
   x = 1 \pm \sqrt{10}
   \]

So, the solutions are:
\[ x = 1 + \sqrt{10} \]
and
\[ x = 1 - \sqrt{10} \]

Would you like more details on any step or have any questions?

Here are some related questions you might be interested in:

1. How do you simplify complex algebraic expressions?
2. Can you explain the steps of the quadratic formula in detail?
3. What are some methods to factor quadratic equations?
4. How do you solve systems of linear equations?
5. Can you demonstrate solving a cubic equation?

**Tip:** When solving quadratic equations, always check if it can be factored easily before applying the quadratic formula. Factoring can sometimes be quicker and simpler.

Learn how to solve the equation -18 + 6x - 7x = -5x + 2x^2 step-by-step with detailed explanations and examples. This problem involves simplifying algebraic expressions and solving quadratic equations. Suitable for high school students.

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