https://file.aiquickdraw.com/mathbot/file/1725485887794nvyrux7z.jpg

The given equation from the image is:

\[
2x^2 + 13x + 17 = 0
\]

This is a quadratic equation, and to solve it for all real solutions, we can use the **quadratic formula**:

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

For the given equation \(2x^2 + 13x + 17 = 0\), we can identify the coefficients:
- \(a = 2\)
- \(b = 13\)
- \(c = 17\)

Let's plug these values into the quadratic formula and calculate the solution. I will compute the discriminant and solutions next.import math

# Coefficients of the quadratic equation
a = 2
b = 13
c = 17

# Calculating the discriminant
discriminant = b**2 - 4*a*c

# Checking if the discriminant is positive, zero, or negative
if discriminant >= 0:
    # Real roots exist, calculate both solutions
    root1 = (-b + math.sqrt(discriminant)) / (2*a)
    root2 = (-b - math.sqrt(discriminant)) / (2*a)
else:
    # Complex roots since discriminant is negative
    real_part = -b / (2*a)
    imaginary_part = math.sqrt(-discriminant) / (2*a)
    root1 = complex(real_part, imaginary_part)
    root2 = complex(real_part, -imaginary_part)

root1, root2The real solutions to the quadratic equation \( 2x^2 + 13x + 17 = 0 \) are:

\[
x_1 \approx -1.81 \quad \text{and} \quad x_2 \approx -4.69
\]

Would you like more details or clarifications on any step?

Here are some follow-up questions that might help deepen your understanding:

1. How is the quadratic formula derived from completing the square?
2. What does the discriminant tell you about the nature of the solutions?
3. How does this approach compare to factoring when solving quadratic equations?
4. Can you graph the quadratic equation and explain the significance of its intercepts?
5. What happens when the discriminant is exactly zero?

**Tip:** The discriminant \(b^2 - 4ac\) is a quick way to determine if a quadratic equation has real or complex solutions—positive discriminant for two real roots, zero for one real root, and negative for complex roots.

Solve the equation for all real solutions in simplest form: 2x^2 + 13x + 17 = 0

Learn how to solve the quadratic equation 2x^2 + 13x + 17 = 0 with a step-by-step guide using the quadratic formula. Understand the discriminant's role in determining the nature of the solutions. Suitable for students in Grades 10-12.

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