How to Factor a Quadratic Expression: Step-by-Step Guide
Quick Answer
To factor the quadratic expression 2h^2 + 9h + 9 completely, first identify if there's a greatest common factor. Then use the 'a·c' method to find two numbers that multiply to 18 and add to 9, which are 3 and 6. Finally, rewrite and factor by grouping.
Factoring quadratic expressions is a fundamental skill in algebra that helps in simplifying equations and solving problems. Let’s take you through the steps to factor the quadratic expression **2h^2 + 9h + 9** completely.
### Step 1: Identify the Greatest Common Factor (GCF)
The first step in factoring is checking for a GCF among the coefficients of the terms. In this case, the coefficients are 2, 9, and 9. The largest number that divides all of these coefficients is 1, indicating that there isn't a GCF larger than 1. Therefore, we can move directly to the next step.
### Step 2: Apply the 'a·c' Method
This quadratic is in the standard form of **ax^2 + bx + c**, where:
- **a = 2** (the coefficient of h^2)
- **b = 9** (the coefficient of h)
- **c = 9** (the constant term)
Next, we calculate **a · c**:
- **2 · 9 = 18**
Now, we need to find two numbers that multiply to 18 and add up to 9 (the value of b). The pairs of factors of 18 are:
- 1 and 18 (sum = 19)
- 2 and 9 (sum = 11)
- **3 and 6 (sum = 9)** ← this pair works!
### Step 3: Rewrite the Quadratic Expression
Once we have the correct pair of numbers, we can rewrite the middle term, 9h, as 3h + 6h. Therefore, we can express the quadratic as:
- **2h^2 + 3h + 6h + 9**
### Step 4: Factor by Grouping
Now, we can group the terms in pairs:
- **(2h^2 + 3h) + (6h + 9)**
Factor out the common factors from each group:
- From the first group, **h(2h + 3)**
- From the second group, we can factor out **3(2h + 3)**
So we can rewrite the expression as:
- **h(2h + 3) + 3(2h + 3)**
### Step 5: Final Factoring
Now, we notice that both terms contain the common factor **(2h + 3)**. Therefore, we can factor that out:
- **(2h + 3)(h + 3)**
### Conclusion
Thus, the completely factored form of the expression **2h^2 + 9h + 9** is **(2h + 3)(h + 3)**. Factoring quadratics is not only crucial for solving equations but also critical in various applications, such as physics and economics, where relationships can be modeled by quadratic equations. Practicing these steps will help you become proficient in factoring and algebra overall.
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