https://file.aiquickdraw.com/mathbot/file/1726748741169tov26oww.jpg

Find the discriminant of each quadratic equation then state the number of real and imaginary solutions

Let's analyze each quadratic equation, find its discriminant, and then determine the number of real or imaginary solutions based on that discriminant.

The discriminant \(\Delta\) of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by:

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

Based on the value of \(\Delta\):
- If \(\Delta > 0\), the equation has **2 distinct real solutions**.
- If \(\Delta = 0\), the equation has **1 real solution** (a repeated root).
- If \(\Delta < 0\), the equation has **2 complex (imaginary) solutions**.

Now, let's calculate the discriminants for each of the quadratic equations:

### 1. \(-2x^2 - 8x - 14 = 6\)
Rewrite the equation as:
\[
-2x^2 - 8x - 14 - 6 = 0
\]
\[
-2x^2 - 8x - 20 = 0
\]
Here, \(a = -2\), \(b = -8\), and \(c = -20\).

Discriminant:
\[
\Delta = (-8)^2 - 4(-2)(-20) = 64 - 160 = -96
\]
Since \(\Delta < 0\), this equation has **2 complex (imaginary) solutions**.

### 2. \(9n^2 - 3n - 8 = -10\)
Rewrite the equation as:
\[
9n^2 - 3n - 8 + 10 = 0
\]
\[
9n^2 - 3n + 2 = 0
\]
Here, \(a = 9\), \(b = -3\), and \(c = 2\).

Discriminant:
\[
\Delta = (-3)^2 - 4(9)(2) = 9 - 72 = -63
\]
Since \(\Delta < 0\), this equation has **2 complex (imaginary) solutions**.

### 3. \(9m^2 + 6m + 6 \geq 5\)
Rewrite the inequality as:
\[
9m^2 + 6m + 6 - 5 \geq 0
\]
\[
9m^2 + 6m + 1 \geq 0
\]
Here, \(a = 9\), \(b = 6\), and \(c = 1\).

Discriminant:
\[
\Delta = 6^2 - 4(9)(1) = 36 - 36 = 0
\]
Since \(\Delta = 0\), this quadratic inequality has **1 real solution** (a repeated root). However, because it's an inequality, the quadratic is **non-negative (always greater or equal to 0)** for all values of \(m\), meaning **all real numbers** will satisfy the inequality.

### 4. \(9b^2 = -8b + 8\)
Rewrite the equation as:
\[
9b^2 + 8b - 8 = 0
\]
Here, \(a = 9\), \(b = 8\), and \(c = -8\).

Discriminant:
\[
\Delta = 8^2 - 4(9)(-8) = 64 + 288 = 352
\]
Since \(\Delta > 0\), this equation has **2 distinct real solutions**.

### Summary of Results:
1. **Equation 1:** 2 complex (imaginary) solutions.
2. **Equation 2:** 2 complex (imaginary) solutions.
3. **Equation 3:** 1 real solution (non-negative for all \(m\)).
4. **Equation 4:** 2 distinct real solutions.

Would you like further details or clarifications on any of these steps?

---

Here are 5 related questions:
1. How do you solve quadratic equations using the quadratic formula?
2. What is the significance of the discriminant in determining solution types?
3. Can a quadratic inequality have complex solutions?
4. How do you solve systems involving quadratic equations?
5. What are some real-world applications of quadratic equations?

**Tip:** When solving a quadratic equation, always simplify the equation first to standard form \(ax^2 + bx + c = 0\) to apply formulas effectively.

Learn how to find the discriminant of quadratic equations and determine the number of real or imaginary solutions. This step-by-step solution covers concepts of quadratic equations and the discriminant formula. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation