https://file.aiquickdraw.com/mathbot/file/1729168209492x508wjgc.jpg

The equation provided in the image is:

\[
3v^2 + 8v = -5
\]

To solve this quadratic equation, we can follow these steps:

### Step 1: Rearrange the equation
First, rewrite the equation in standard quadratic form:

\[
3v^2 + 8v + 5 = 0
\]

### Step 2: Use the quadratic formula
The quadratic formula is:

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

Here, \(a = 3\), \(b = 8\), and \(c = 5\).

### Step 3: Calculate the discriminant
The discriminant is:

\[
\Delta = b^2 - 4ac = 8^2 - 4(3)(5) = 64 - 60 = 4
\]

Since the discriminant is positive, there are two real solutions.

### Step 4: Solve for \(v\)
Now substitute the values into the quadratic formula:

\[
v = \frac{-8 \pm \sqrt{4}}{2(3)} = \frac{-8 \pm 2}{6}
\]

This gives two solutions:

1. \(v_1 = \frac{-8 + 2}{6} = \frac{-6}{6} = -1\)
2. \(v_2 = \frac{-8 - 2}{6} = \frac{-10}{6} = \frac{-5}{3}\)

### Final solutions:
The solutions are:

\[
v = -1, \frac{-5}{3}
\]

Let me know if you'd like further details or explanations! 

Here are some questions to expand on this:
1. What is the quadratic formula and when do we use it?
2. How do we interpret the discriminant?
3. Can a quadratic equation have no real solution?
4. What if the discriminant is zero? How many solutions are there?
5. What is the difference between solving quadratics by factoring and using the quadratic formula?

**Tip:** Always check the discriminant before solving a quadratic equation to know how many real solutions to expect.

Solve for v in the equation 3v^2 + 8v = -5.

Learn how to solve the quadratic equation 3v² + 8v + 5 = 0 using the quadratic formula. This step-by-step solution covers algebraic concepts and discriminant analysis, suitable for students in Grades 9-11.

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