solve the equation

Amanita, let's focus on the equation inside the green rectangle:
$3x^2 - 10x + 8 = 0$

Here’s the step-by-step plan to solve this quadratic equation:

Step 1: Identify the coefficients.
The equation is in the standard quadratic form $ax^2 + bx + c = 0$, where:

• $a = 3$
• $b = -10$
• $c = 8$

Step 2: Use the quadratic formula.
The quadratic formula to solve $ax^2 + bx + c = 0$ is:

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

We will substitute the values of $a$, $b$, and $c$ into this formula.

Step 3: Calculate the discriminant.
The discriminant is $b^2 - 4ac$. This will help us determine the number and type of solutions.

Step 4: Find the solutions for $x$ using the quadratic formula.

Now, let's execute these steps.

First, let's solve the equation $3x^2 - 10x + 8 = 0$ for $x$. Amanita, here’s how we solve the equation $3x^2 - 10x + 8 = 0$:

1. We identify the coefficients: $a = 3$, $b = -10$, and $c = 8$.
2. We use the quadratic formula:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

3. Calculating inside the square root:

$(-10)^2 - 4 \cdot 3 \cdot 8 = 100 - 96 = 4$

4. Substitute back:

$x = \frac{10 \pm \sqrt{4}}{6}$

$x = \frac{10 \pm 2}{6}$

5. This gives us two solutions:

$x_1 = \frac{10 + 2}{6} = \frac{12}{6} = 2$

$x_2 = \frac{10 - 2}{6} = \frac{8}{6} = \frac{4}{3}$

So, the solutions to the equation are:

$x = 2 \quad \text{and} \quad x = \frac{4}{3}$