https://file.aiquickdraw.com/mathbot/file/17269970484409r8aw8t9.jpg

The image shows the following equations and expressions:

1. The first equation:
\[
\frac{2x^2 - 5x + 3}{5} = 3
\]

2. A second equation:
\[
2x^2 - 5x + 3 = 3
\]

3. The text "x_1, x_2" followed by an expression with a green mark obscuring part of the word, which likely says "make," and then the expression:
\[
x_1 - x_2 =
\]

### Let's start by solving the first equation:

\[
\frac{2x^2 - 5x + 3}{5} = 3
\]

Multiply both sides by 5 to eliminate the fraction:
\[
2x^2 - 5x + 3 = 15
\]

Next, subtract 15 from both sides to set the equation to zero:
\[
2x^2 - 5x + 3 - 15 = 0
\]
\[
2x^2 - 5x - 12 = 0
\]

Now, we have a quadratic equation:
\[
2x^2 - 5x - 12 = 0
\]

### Solving the quadratic equation:
We can use the quadratic formula to find \(x_1\) and \(x_2\):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = -5\), and \(c = -12\).

Let's calculate the solutions.The solutions to the quadratic equation \(2x^2 - 5x - 12 = 0\) are:

\[
x_1 = 4 \quad \text{and} \quad x_2 = -1.5
\]

### Next step:
The expression at the end, \(x_1 - x_2\), asks for the difference between these two solutions:
\[
x_1 - x_2 = 4 - (-1.5) = 4 + 1.5 = 5.5
\]

Thus, \(x_1 - x_2 = 5.5\).

Would you like more details, or do you have any questions?

Here are five related questions you could explore:
1. How does the discriminant help determine the nature of the roots in a quadratic equation?
2. Can we solve quadratic equations using factoring instead of the quadratic formula?
3. How do we complete the square to solve a quadratic equation?
4. What is the graphical interpretation of the roots of a quadratic equation?
5. How does changing the coefficients \(a\), \(b\), or \(c\) in a quadratic equation affect its solutions?

**Tip:** Always check the discriminant before using the quadratic formula to confirm if the roots are real, imaginary, or repeated.

Solve the quadratic equation: (2x^2 - 5x + 3) / 5 = 3. Also, find the difference between x1 and x2.

Learn how to solve the quadratic equation (2x^2 - 5x + 3) / 5 = 3 and find the difference between its roots x1 and x2. This step-by-step solution covers key algebraic concepts and demonstrates the use of the quadratic formula.

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