Mastering Factoring x^2 + bx + c: Advanced Examples

📚 Quick Study Guide

• 🔢 Factoring Basics: We're looking to express $x^2 + bx + c$ as a product of two binomials: $(x + r)(x + s)$.
• ➕ Sum and Product: Find two numbers, $r$ and $s$, such that $r + s = b$ (the sum) and $r \cdot s = c$ (the product).
• 💡 Advanced Tip: When $c$ is negative, one of $r$ and $s$ is positive and the other is negative. When $c$ is positive, both $r$ and $s$ have the same sign as $b$.
• 🧮 Trial and Error: Sometimes, you might need to try a few different combinations of $r$ and $s$ to find the correct pair.
• ✍️ Check Your Work: Always multiply the factored form $(x + r)(x + s)$ back out to make sure it equals the original expression $x^2 + bx + c$.

Practice Quiz

1. Question 1: Factor $x^2 + 8x + 15$.
   1. (x + 2)(x + 6)
   2. (x + 3)(x + 5)
   3. (x + 1)(x + 15)
   4. (x + 4)(x + 4)
2. Question 2: Factor $x^2 - 5x + 6$.
   1. (x - 2)(x - 3)
   2. (x + 2)(x + 3)
   3. (x - 1)(x - 6)
   4. (x + 1)(x - 6)
3. Question 3: Factor $x^2 + 2x - 8$.
   1. (x + 4)(x - 2)
   2. (x - 4)(x + 2)
   3. (x + 1)(x - 8)
   4. (x - 1)(x + 8)
4. Question 4: Factor $x^2 - 7x - 18$.
   1. (x - 9)(x + 2)
   2. (x + 9)(x - 2)
   3. (x - 6)(x - 3)
   4. (x + 6)(x + 3)
5. Question 5: Factor $x^2 + 11x + 30$.
   1. (x + 5)(x + 6)
   2. (x + 2)(x + 15)
   3. (x + 3)(x + 10)
   4. (x + 1)(x + 30)
6. Question 6: Factor $x^2 - 4x - 21$.
   1. (x - 7)(x + 3)
   2. (x + 7)(x - 3)
   3. (x - 1)(x + 21)
   4. (x + 1)(x - 21)
7. Question 7: Factor $x^2 - 16x + 63$.
   1. (x - 7)(x - 9)
   2. (x + 7)(x + 9)
   3. (x - 3)(x - 21)
   4. (x + 3)(x + 21)