To solve the quadratic equation \( 3z^2 + 11z - 14 = 0 \), we will use the quadratic formula:

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

Where:
- \(a = 3\),
- \(b = 11\),
- \(c = -14\).

### Step 1: Calculate the discriminant

The discriminant is the part under the square root in the quadratic formula, given by:

\[
\Delta = b^2 - 4ac
\]

Substitute the values of \(a\), \(b\), and \(c\):

\[
\Delta = 11^2 - 4(3)(-14) = 121 + 168 = 289
\]

### Step 2: Use the quadratic formula

Now substitute the values into the quadratic formula:

\[
z = \frac{-11 \pm \sqrt{289}}{2(3)} = \frac{-11 \pm 17}{6}
\]

### Step 3: Solve for the two possible values of \(z\)

1. \( z_1 = \frac{-11 + 17}{6} = \frac{6}{6} = 1 \)
2. \( z_2 = \frac{-11 - 17}{6} = \frac{-28}{6} = -\frac{14}{3} \)

### Final Solution:

The two solutions are:

\[
z_1 = 1, \quad z_2 = -\frac{14}{3}
\]

Would you like more details or have any questions?

---

Here are some related questions you might find helpful:

1. How can we check the solutions to a quadratic equation?
2. What is the significance of the discriminant in a quadratic equation?
3. How can we factorize quadratic equations manually?
4. What is the difference between real and complex roots?
5. How is the quadratic formula derived?

**Tip:** The discriminant helps to determine the nature of the roots: if it's positive, the equation has two real roots!

Explore how to solve the quadratic equation 3z^2 + 11z - 14 using the quadratic formula. Follow a step-by-step guide to calculate the discriminant and find the two possible solutions. This problem is ideal for students in Grades 9-10 learning key algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation