solve the quadratic equation by completing the squares, leave it in surd form

9-2x-3x^2=0

To solve the quadratic equation \( 9 - 2x - 3x^2 = 0 \) by completing the square, follow these steps:

### Step 1: Rewrite the equation in standard form
The given equation is:

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

First, rearrange the terms to place the quadratic term first:

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

Now, divide the entire equation by \(-1\) to simplify:

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

### Step 2: Move the constant term to the right side
\[
3x^2 + 2x = 9
\]

### Step 3: Factor out the coefficient of \(x^2\)
Factor out the 3 from the left side:

\[
3 \left( x^2 + \frac{2}{3}x \right) = 9
\]

### Step 4: Complete the square
To complete the square inside the parentheses, take the coefficient of \(x\), which is \(\frac{2}{3}\), divide it by 2, and then square it:

\[
\left(\frac{2}{3} \times \frac{1}{2}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}
\]

Add and subtract \(\frac{1}{9}\) inside the parentheses:

\[
3 \left( x^2 + \frac{2}{3}x + \frac{1}{9} - \frac{1}{9} \right) = 9
\]

This simplifies to:

\[
3 \left( \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} \right) = 9
\]

### Step 5: Simplify and solve for \(x\)
Distribute the 3:

\[
3\left(x + \frac{1}{3}\right)^2 - 3\left(\frac{1}{9}\right) = 9
\]

\[
3\left(x + \frac{1}{3}\right)^2 - \frac{1}{3} = 9
\]

Add \(\frac{1}{3}\) to both sides:

\[
3\left(x + \frac{1}{3}\right)^2 = 9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}
\]

Divide by 3:

\[
\left(x + \frac{1}{3}\right)^2 = \frac{28}{9}
\]

### Step 6: Take the square root of both sides
\[
x + \frac{1}{3} = \pm \sqrt{\frac{28}{9}} = \pm \frac{\sqrt{28}}{3}
\]

### Step 7: Solve for \(x\)
Finally, subtract \(\frac{1}{3}\) from both sides:

\[
x = -\frac{1}{3} \pm \frac{\sqrt{28}}{3}
\]

Simplify the square root:

\[
x = \frac{-1 \pm 2\sqrt{7}}{3}
\]

So, the solutions are:

\[
x = \frac{-1 + 2\sqrt{7}}{3} \quad \text{or} \quad x = \frac{-1 - 2\sqrt{7}}{3}
\]

These are the solutions in surd form.

Would you like to go over any part of the process in more detail, or do you have any questions?

### Related Questions
1. How do you solve quadratic equations using the quadratic formula?
2. What is the difference between completing the square and factoring?
3. How do you derive the quadratic formula from completing the square?
4. How can you identify when completing the square is the best method to use?
5. Can all quadratic equations be solved by completing the square?

### Tip
When completing the square, always remember to balance the equation by adding and subtracting the same value, and simplify the square root carefully for an exact solution.

solve the quadratic equation by completing the squares, leave it in surd form

9 - 2x - 3x^2 = 0

Learn how to solve the quadratic equation 9 - 2x - 3x^2 = 0 using the completing the square method, leaving the answer in surd form. This step-by-step solution covers essential algebraic techniques and is ideal for students in Grades 9-10.

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