Could you explain step 3:

Step 3: Analyze the Discriminant

Certainly! Analyzing the discriminant is a crucial step in solving quadratic equations. Here’s an explanation of what the discriminant is and how to analyze it:

### What is the Discriminant?
The discriminant is a component of the quadratic formula used to solve quadratic equations of the form:

\[
ax^2 + bx + c = 0
\]

The quadratic formula is:

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

In this formula, the expression under the square root, \(b^2 - 4ac\), is called the **discriminant**.

### How to Analyze the Discriminant?
The value of the discriminant determines the nature and number of the roots (solutions) of the quadratic equation:

1. **Positive Discriminant (\(b^2 - 4ac > 0\)):**
   - If the discriminant is positive, the quadratic equation has **two distinct real roots**.
   - This means the graph of the quadratic function intersects the x-axis at two points.

2. **Zero Discriminant (\(b^2 - 4ac = 0\)):**
   - If the discriminant is zero, the quadratic equation has **exactly one real root** (or a repeated real root).
   - This means the graph of the quadratic function touches the x-axis at exactly one point, known as a double root.

3. **Negative Discriminant (\(b^2 - 4ac < 0\)):**
   - If the discriminant is negative, the quadratic equation has **two complex roots** (conjugates).
   - This means the graph of the quadratic function does not intersect the x-axis.

### Summary of the Steps to Analyze the Discriminant:
1. **Calculate the discriminant** using the formula \(b^2 - 4ac\).
2. **Determine the sign** (positive, zero, or negative) of the discriminant.
3. **Interpret the result** to find out how many and what type of roots the quadratic equation has.

Would you like more details or have any other questions?  

Here are 5 related questions that might interest you:
1. How does the discriminant relate to the nature of the roots?
2. Can you show an example of finding the discriminant and solving a quadratic equation?
3. What are complex roots, and how do they appear in quadratic equations?
4. How does the discriminant affect the graph of a quadratic function?
5. Can the discriminant be used in equations that are not quadratic?

**Tip:** Always check the sign of the discriminant before solving the quadratic equation, as it guides you in choosing the correct method to find the roots.

Understand how to analyze the discriminant in quadratic equations with a detailed step-by-step explanation. Learn about the quadratic formula, discriminant calculation, and its implications on the nature of roots. Suitable for high school students and those preparing for standardized tests.

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