https://file.aiquickdraw.com/mathbot/file/1727031644176omn452bk.jpg

We are given the quadratic equation:

\[
3y^2 + 5y - 12 = 0
\]

We will solve this using the **quadratic formula**, which is:

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

For the given equation \(3y^2 + 5y - 12 = 0\), we have the coefficients:

- \(a = 3\)
- \(b = 5\)
- \(c = -12\)

### Step 1: Calculate the discriminant
The discriminant is:

\[
\Delta = b^2 - 4ac = 5^2 - 4(3)(-12) = 25 + 144 = 169
\]

### Step 2: Apply the quadratic formula
Now, we substitute into the quadratic formula:

\[
y = \frac{-5 \pm \sqrt{169}}{2(3)} = \frac{-5 \pm 13}{6}
\]

### Step 3: Find the roots
We have two cases for the roots:

1. \(\frac{-5 + 13}{6} = \frac{8}{6} = \frac{4}{3}\)
2. \(\frac{-5 - 13}{6} = \frac{-18}{6} = -3\)

Thus, the roots are \(y = \frac{4}{3}\) and \(y = -3\).

### Answer:
The correct answer is the first option: \(\left\{ \frac{4}{3}, -3 \right\}\).

Would you like further details on this solution or have any questions?

Here are some related questions:
1. How do we determine if a quadratic equation has real roots?
2. What does the discriminant tell us about the nature of roots?
3. How would the roots change if the discriminant was negative?
4. Can the quadratic formula be used to solve any quadratic equation?
5. How does the value of 'a' affect the shape of a parabola?

**Tip:** Always check the discriminant before solving with the quadratic formula to know the nature of the roots.

What are the roots of the quadratic equation 3y^2 + 5y - 12 = 0?

Learn how to solve the quadratic equation 3y^2 + 5y - 12 = 0 using the quadratic formula. This example covers key concepts like discriminant calculation and applying the quadratic formula step by step. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation