To solve the equation \(10 - 6x - 9x^2 = -9x\), we first need to rearrange it into the standard quadratic form, \(ax^2 + bx + c = 0\).

### Step 1: Move all terms to one side of the equation

\[
10 - 6x - 9x^2 + 9x = 0
\]

Combine like terms:

\[
10 + 3x - 9x^2 = 0
\]

Now, the quadratic equation is in standard form:

\[
-9x^2 + 3x + 10 = 0
\]

Here, \(a = -9\), \(b = 3\), and \(c = 10\).

### Step 2: Apply the quadratic formula

The quadratic formula is:

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

Substitute \(a = -9\), \(b = 3\), and \(c = 10\):

\[
x = \frac{-3 \pm \sqrt{(3)^2 - 4(-9)(10)}}{2(-9)}
\]

### Step 3: Calculate the discriminant

Compute the discriminant (\(b^2 - 4ac\)):

\[
(3)^2 - 4(-9)(10) = 9 + 360 = 369
\]

### Step 4: Solve for \(x\)

Substitute the discriminant back into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{369}}{-18}
\]

Simplify the expression:

\[
x_1 = \frac{-3 + \sqrt{369}}{-18}, \quad x_2 = \frac{-3 - \sqrt{369}}{-18}
\]

Simplify further by multiplying numerator and denominator by \(-1\):

\[
x_1 = \frac{3 - \sqrt{369}}{18}, \quad x_2 = \frac{3 + \sqrt{369}}{18}
\]

### Solutions

The solutions for \(x\) are:

\[
x_1 = \frac{3 - \sqrt{369}}{18}, \quad x_2 = \frac{3 + \sqrt{369}}{18}
\]

These are the two roots of the quadratic equation.

Would you like a more detailed explanation of any step or have any questions?

Here are some related questions to consider:

1. How can you verify the solutions of a quadratic equation?
2. What does the graph of this quadratic equation look like?
3. How do the signs of \(a\), \(b\), and \(c\) affect the parabola's direction and shape?
4. How can factoring be used to solve quadratic equations?
5. When is it more efficient to use the quadratic formula versus factoring?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before applying the quadratic formula to understand the nature of the roots (real or complex).

Learn how to solve the quadratic equation 10 - 6x - 9x^2 = -9x with a detailed step-by-step solution. This problem involves rearranging the equation, applying the quadratic formula, and calculating roots. Suitable for students in Grades 10-12.

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