To solve the quadratic equation \( x^2 + ax + b = 0 \), you can use the quadratic formula. The quadratic formula is derived from the general form of a quadratic equation \( ax^2 + bx + c = 0 \), but in your case, \( a = 1 \), \( b = a \), and \( c = b \). Here's the step-by-step process:

1. **Identify the coefficients**: In the equation \( x^2 + ax + b = 0 \), the coefficients are:
   - \( A = 1 \)
   - \( B = a \)
   - \( C = b \)

2. **Write down the quadratic formula**: The quadratic formula is given by:
   \[
   x = \frac{{-B \pm \sqrt{{B^2 - 4AC}}}}{2A}
   \]

3. **Plug in the coefficients**: Substitute \( A = 1 \), \( B = a \), and \( C = b \) into the formula:
   \[
   x = \frac{{-a \pm \sqrt{{a^2 - 4 \cdot 1 \cdot b}}}}{2 \cdot 1}
   \]
   Simplifying further:
   \[
   x = \frac{{-a \pm \sqrt{{a^2 - 4b}}}}{2}
   \]

4. **Calculate the discriminant**: The discriminant is the part under the square root, \( a^2 - 4b \). The value of the discriminant determines the nature of the roots:
   - If \( a^2 - 4b > 0 \), there are two distinct real roots.
   - If \( a^2 - 4b = 0 \), there is one real root (a repeated root).
   - If \( a^2 - 4b < 0 \), there are two complex roots.

5. **Find the roots**: Using the quadratic formula, compute the roots based on the discriminant:
   - For two distinct real roots:
     \[
     x_1 = \frac{{-a + \sqrt{{a^2 - 4b}}}}{2}
     \]
     \[
     x_2 = \frac{{-a - \sqrt{{a^2 - 4b}}}}{2}
     \]
   - For one real root:
     \[
     x = \frac{{-a}}{2}
     \]
   - For two complex roots:
     \[
     x_1 = \frac{{-a + \sqrt{{a^2 - 4b}}}}{2}
     \]
     \[
     x_2 = \frac{{-a - \sqrt{{a^2 - 4b}}}}{2}
     \]

By following these steps, you can solve the quadratic equation \( x^2 + ax + b = 0 \).

Would you like more details on any part of this process or have any questions?

Here are 8 related questions that might expand your understanding:
1. How is the quadratic formula derived?
2. What are the conditions for a quadratic equation to have real roots?
3. How do you complete the square for a quadratic equation?
4. Can you solve \( x^2 + 4x + 4 = 0 \) using the quadratic formula?
5. What are the properties of the discriminant in a quadratic equation?
6. How do you graph a quadratic equation?
7. How do you solve a quadratic equation by factoring?
8. What are the applications of quadratic equations in real-life scenarios?

**Tip:** Always check the discriminant \( a^2 - 4b \) before solving the quadratic equation to understand the nature of the roots you will get.

Learn how to solve the quadratic equation x^2 + ax + b = 0 with a detailed step-by-step solution using the quadratic formula. Understand the nature of roots based on the discriminant value. Suitable for Grades 10-12.

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