When solving simple trinomials, the most common form you'll encounter is:

$ax^2 + bx + c = 0$

Steps for Solving Simple Trinomials:

1. Factor the trinomial:

   • You want to factor it into two binomials of the form: $(px + q)(rx + s)$
   • Look for two numbers that multiply to give $a \times c$ (the product of the coefficient of $x^2$ and the constant term) and add up to $b$ (the coefficient of $x$).

2. Find the factors of $a \times c$:

   • If $a = 1$, you only need to focus on factors of $c$.
   • If $a \neq 1$, find pairs of numbers that multiply to $a \times c$ and check if their sum is $b$.

3. Split the middle term:

   • Rewrite the middle term $bx$ as the sum of two terms based on the factors found in step 2.

4. Factor by grouping:

   • Group the terms and factor out the greatest common factor (GCF) from each group.

5. Solve for $x$ (if you are solving an equation):

   • Set each factor equal to 0 and solve for $x$.

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Example 1: Factor the trinomial

Consider the trinomial:

$x^2 + 5x + 6$

• Step 1: Look for two numbers that multiply to $1 \times 6 = 6$ and add to 5.

  • The numbers are 2 and 3 because $2 \times 3 = 6$ and $2 + 3 = 5$.

• Step 2: Split the middle term using 2 and 3: $x^2 + 2x + 3x + 6$

• Step 3: Factor by grouping: $(x^2 + 2x) + (3x + 6)$ Factor out the GCF from each group: $x(x + 2) + 3(x + 2)$

• Step 4: Factor out the common binomial factor: $(x + 2)(x + 3)$

So, the factored form of the trinomial is: $(x + 2)(x + 3)$

Example 2: Solve the trinomial equation

Now, let's solve the equation: $x^2 + 5x + 6 = 0$

• From the factorization above: $(x + 2)(x + 3) = 0$

• Set each factor equal to zero: $x + 2 = 0 \quad \text{or} \quad x + 3 = 0$ Solving these: $x = -2 \quad \text{or} \quad x = -3$

Thus, the solutions are $x = -2$ and $x = -3$.

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Would you like more examples or any specific details on a particular step?

Related Questions:

1. How do I factor trinomials when $a \neq 1$?
2. What is the difference between factoring and solving a trinomial equation?
3. Can you explain factoring with the AC method?
4. How can I check if my factorization is correct?
5. How does factoring apply to quadratic functions?

Tip: Always double-check your factor pairs to ensure their sum matches the middle term.