how do I factor 12xsubscript2 + 27x + 6, when 6 cannot factor into 27

To factor the quadratic expression \(12x^2 + 27x + 6\), we can use the method of factoring by grouping. Here's how to do it step by step:

### Step 1: Identify the coefficients
We have the quadratic in the form:
\[
ax^2 + bx + c
\]
Where:
- \(a = 12\)
- \(b = 27\)
- \(c = 6\)

### Step 2: Multiply \(a\) and \(c\)
Multiply the first coefficient \(a\) (which is 12) by the last constant \(c\) (which is 6):
\[
12 \times 6 = 72
\]
Now, we need to find two numbers that multiply to give 72 and add to give the middle coefficient \(b = 27\).

### Step 3: Find the pair of numbers
The pair of numbers that multiply to 72 and add to 27 are 24 and 3:
\[
24 \times 3 = 72 \quad \text{and} \quad 24 + 3 = 27
\]

### Step 4: Rewrite the middle term
Now, rewrite the quadratic expression by splitting the middle term (27x) into two terms using the numbers we found (24 and 3):
\[
12x^2 + 24x + 3x + 6
\]

### Step 5: Factor by grouping
Group the terms in pairs:
\[
(12x^2 + 24x) + (3x + 6)
\]
Now factor out the greatest common factor (GCF) from each group:
\[
12x(x + 2) + 3(x + 2)
\]

### Step 6: Factor out the common binomial
Both terms have a common binomial factor of \((x + 2)\), so we can factor that out:
\[
(12x + 3)(x + 2)
\]

### Step 7: Simplify
Notice that we can factor further:
\[
12x + 3 = 3(4x + 1)
\]
So the fully factored form is:
\[
3(4x + 1)(x + 2)
\]

Thus, the factored form of \(12x^2 + 27x + 6\) is:
\[
3(4x + 1)(x + 2)
\]

### Do you have any questions or would you like further details?

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Here are 5 related questions to expand on this concept:
1. How would you factor a quadratic where the middle term is negative, like \(12x^2 - 27x + 6\)?
2. Can you factor a quadratic where no two integers multiply to \(a \times c\) and add to \(b\)?
3. What happens if the quadratic expression has no real factors (i.e., the discriminant is negative)?
4. How do you complete the square for the expression \(12x^2 + 27x + 6\)?
5. Can you apply the quadratic formula to solve \(12x^2 + 27x + 6 = 0\)?

**Tip**: Always check for a greatest common factor (GCF) before starting the factoring process—it simplifies the expression early!

how do I factor 12x^2 + 27x + 6, when 6 cannot factor into 27

Learn how to factor the quadratic expression 12x^2 + 27x + 6 using the method of factoring by grouping. This detailed step-by-step solution covers essential algebraic concepts and formulas, suitable for students in Grades 9-12.

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