Factoring Quadratics

Factoring quadratics involves expressing a quadratic expression in the form $ax^{2} + bx + c$ as a product of two binomials. The goal is to find two numbers that multiply to $ac$ and add to $b$.

Example

Let's factor the quadratic expression $2x^{2} + 7x + 3$.

1. Identify $a$, $b$, and $c$:
   Here, $a = 2$, $b = 7$, and $c = 3$.

2. Calculate $ac$:

   $$ac = 2 \cdot 3 = 6$$

3. Find two numbers that multiply to $6$ and add to $7$:
   The numbers are $6$ and $1$.

4. Rewrite the middle term using these numbers:

   $$2x^{2} + 6x + 1x + 3$$

5. Group the terms:

   $$(2x^{2} + 6x) + (1x + 3)$$

6. Factor out the common factors in each group:

   $$2x(x + 3) + 1(x + 3)$$

7. Factor out the common binomial:

   $$(2x + 1)(x + 3)$$

Thus, the factored form of $2x^{2} + 7x + 3$ is $(2x + 1)(x + 3)$.

Key Questions

• What are the steps to factor $x^{2} + 5x + 6$?
• How do you identify $a$, $b$, and $c$ in a quadratic expression?
• Can all quadratics be factored using integers?
• What if the quadratic does not factor neatly? What methods can be applied?
• How do you check if your factoring is correct?

Keywords

• Quadratic
• Factor
• Binomial
• Roots
• Polynomial