Detailed Examples of Factoring Quadratic Expressions $ax^2+bx+c$ When $a \ne 1$

📚 Quick Study Guide

• 🔍 When factoring a quadratic expression in the form $ax^2 + bx + c$ where $a \ne 1$, the goal is to find two binomials that, when multiplied, give you the original quadratic.
• 🔢 The 'ac' method involves multiplying $a$ and $c$, then finding two numbers that multiply to this product and add up to $b$.
• ➗ Once you find these two numbers (let's call them $m$ and $n$), rewrite the middle term $bx$ as $mx + nx$.
• ➗ Now, factor by grouping. Group the first two terms and the last two terms, and factor out the greatest common factor (GCF) from each group.
• 🤝 If done correctly, you should have a common binomial factor. Factor out this common binomial.
• ✍️ The result will be two binomials that represent the factored form of the original quadratic expression.
• 💡 Always check your answer by multiplying the two binomials to ensure you get back the original quadratic expression.

🧪 Practice Quiz

1. What is the factored form of $2x^2 + 7x + 3$?
1. $(2x + 1)(x + 3)$
2. $(2x + 3)(x + 1)$
3. $(x + 1)(x + 3)$
4. $(2x - 1)(x - 3)$
2. Factor $3x^2 + 10x + 8$.
1. $(3x + 4)(x + 2)$
2. $(3x + 2)(x + 4)$
3. $(x + 2)(x + 4)$
4. $(3x - 4)(x - 2)$
3. What are the factors of $4x^2 - 8x - 5$?
1. $(2x - 5)(2x + 1)$
2. $(2x + 5)(2x - 1)$
3. $(4x - 5)(x + 1)$
4. $(4x + 1)(x - 5)$
4. Factor the expression $6x^2 + 5x - 4$.
1. $(3x + 4)(2x - 1)$
2. $(3x - 4)(2x + 1)$
3. $(6x - 1)(x + 4)$
4. $(6x + 1)(x - 4)$
5. Which of the following is the factored form of $5x^2 - 13x + 6$?
1. $(5x - 3)(x - 2)$
2. $(5x - 2)(x - 3)$
3. $(5x + 3)(x + 2)$
4. $(5x + 2)(x + 3)$
6. Factor $7x^2 + 22x + 3$.
1. $(7x + 1)(x + 3)$
2. $(7x + 3)(x + 1)$
3. $(x + 1)(x + 3)$
4. $(7x - 1)(x - 3)$
7. What is the factored form of $8x^2 - 14x + 3$?
1. $(4x - 1)(2x - 3)$
2. $(4x - 3)(2x - 1)$
3. $(8x - 1)(x - 3)$
4. $(8x - 3)(x - 1)$