This a solving Q.E by Factoring ax²+bx+c

When solving a quadratic equation of the form \( ax^2 + bx + c = 0 \) by factoring, the goal is to express the quadratic expression as a product of two binomials. Here's a step-by-step guide:

### 1. **Identify the Coefficients:**
   The given quadratic equation is:
   \[
   ax^2 + bx + c = 0
   \]
   For the equation you provided:
   \[
   x^2 + 20x + 100 = 0
   \]
   Here, \( a = 1 \), \( b = 20 \), and \( c = 100 \).

### 2. **Check for Factoring Possibility:**
   Factoring is possible if you can find two numbers that multiply to \( ac \) (in this case \( 1 \times 100 = 100 \)) and add to \( b \) (in this case \( 20 \)).

   The numbers that multiply to 100 and add to 20 are \( 10 \) and \( 10 \).

### 3. **Rewrite the Middle Term:**
   Rewrite the middle term using the two numbers found:
   \[
   x^2 + 10x + 10x + 100 = 0
   \]

### 4. **Factor by Grouping:**
   Group the terms to factor them:
   \[
   (x^2 + 10x) + (10x + 100) = 0
   \]
   Factor out the common factors from each group:
   \[
   x(x + 10) + 10(x + 10) = 0
   \]

### 5. **Factor the Common Binomial:**
   Now, factor out the common binomial \( (x + 10) \):
   \[
   (x + 10)(x + 10) = 0
   \]
   This simplifies to:
   \[
   (x + 10)^2 = 0
   \]

### 6. **Solve for \( x \):**
   Set each factor equal to zero:
   \[
   x + 10 = 0
   \]
   So, the solution is:
   \[
   x = -10
   \]

### Summary:
The solution to the quadratic equation \( x^2 + 20x + 100 = 0 \) by factoring is \( x = -10 \).

Would you like further details or have any questions?

Here are some related questions to explore:
1. How can you factor more complex quadratic equations?
2. What are some common mistakes to avoid when factoring quadratic equations?
3. How do you factor quadratic equations when \( a \) is not equal to 1?
4. What are the differences between factoring and using the quadratic formula?
5. Can all quadratic equations be factored? Why or why not?

**Tip:** When factoring, always check your factors by expanding them back to ensure they match the original quadratic expression.

Learn how to solve quadratic equations by factoring using a step-by-step example. This detailed solution covers the quadratic equation ax^2 + bx + c = 0, demonstrating the process with an example problem. Suitable for high school students studying algebra.

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