Let's solve the quadratic equation given in the image using the quadratic formula.

The equation is:

$9x^2 = 4x + 2$

First, we need to rearrange it into standard form:

$9x^2 - 4x - 2 = 0$

In the quadratic equation $ax^2 + bx + c = 0$, we have:

• $a = 9$
• $b = -4$
• $c = -2$

Now we can apply the quadratic formula:

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

Step-by-Step Solution:

1. Substitute $a$, $b$, and $c$ into the formula:

   $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 9 \cdot (-2)}}{2 \cdot 9}$

2. Simplify inside the square root:

   $x = \frac{4 \pm \sqrt{16 + 72}}{18}$

   $x = \frac{4 \pm \sqrt{88}}{18}$

3. Simplify $\sqrt{88}$:

   $\sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22}$

   So the equation becomes:

   $x = \frac{4 \pm 2\sqrt{22}}{18}$

4. Simplify the fraction by dividing both the numerator and the denominator by 2:

   $x = \frac{2 \pm \sqrt{22}}{9}$

Solution Set:

The solutions are:

$x = \frac{2 + \sqrt{22}}{9}, \quad x = \frac{2 - \sqrt{22}}{9}$

The solution set is therefore:

$\left\{ \frac{2 + \sqrt{22}}{9}, \frac{2 - \sqrt{22}}{9} \right\}$

Would you like more details on any of these steps?

Here are some related questions for further exploration:

1. How is the discriminant in the quadratic formula used to determine the nature of the roots?
2. What are other methods for solving quadratic equations besides the quadratic formula?
3. How can you factor a quadratic equation if it is factorable?
4. What does it mean when a quadratic equation has complex roots?
5. How can completing the square be used to solve quadratic equations?

Tip: Always check if a quadratic equation can be factored first before using the quadratic formula—it can sometimes simplify the solution process!